Phase field group:

Leader: Prof. László Gránásy (DSc) 
Members: Tamás Pusztai (PhD), György Tegze (PhD), Gyula I. Tóth (PhD), László Rátkai (PhD student), Frigyes Podmaniczky (PhD student), Bálint Korbuly (PhD student)

 

Previous members: László Környei (PhD); Tamás Börzsönyi (PhD), Attila Szállás (PhD)

Latest results:

Dendrites Regularized by Spatially Homogeneous Time-Periodic Forcing

T. Börzsönyi, T. Tóth-Katona, Á. Buka, and L. Gránásy

Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences, P.O.B. 49, H-1525 Budapest, Hungary

 

The effect of spatially homogeneous time-periodic external forcing on dendritic solidification has been studied by phase-field modeling and experiments on liquid crystal. It is shown that the frequency of dendritic sidebranching can be tuned by oscillating pressure or heating. The main parameters that influence this phenomenon are identified. [Phys. Rev. Lett. 83, 2853-2856 (1999)].

Nucleation and Bulk Crystallization in Binary Phase Field Theory

László Gránásy,¹ Tamás Börzsönyi,^1,2 and Tamás Pusztai¹

¹Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary
²Groupe de Physique des Solides, CNRS UMR 75-88, Universités Paris VI at VII, Tour 23, 2 place Jussieu, 75251 Paris Cedex 05, France

 

We present a phase field theory for binary crystal nucleation. In the one-component limit, quantitative agreement is achieved with computer simulations (Lennard-Jones system) and experiments (ice-water system) using model parameters evaluated from the free energy and thickness of the interface. The critical undercoolings predicted for Cu-Ni alloys accord with the measurements, and indicate homogeneous nucleation. The Kolmogorov exponents deduced for dendritic solidification and for soft impingement of particles via diffusion fields are consistent with experiment. [Phys. Rev. Lett. 88, 206105 (2002)].

Crystal nucleation and growth in binary phase-field theory

László Gránásy,¹ Tamás Börzsönyi,^1,2 and Tamás Pusztai¹

¹Research Institute for Solid State Physics and Optics, POB 49, H-1525 Budapest, Hungary
²Groupe de Physique des Solides, CNRS UMR 75-88, Universités Paris VI at VII, Tour 23, 2 place Jussieu,75251, Paris Cedex 05, France

Nucleation and growth in unary and binary systems is investigated in the framework of the phase-field theory. Evaluating the model parameters from the interfacial free energy and interface thickness, a quantitative agreement is found with computer simulations and experiments on the ice water system. The critical undercoolings predicted for a simple binary system are close to experiment. Phase-field simulations for isotropic and anisotropic systems show that due to the interacting diffusion fields the Avrami Kolmogorov exponent varies with transformed fraction and initial concentration.  [Journal of Cryst. Growth, 237-239, 1813 (2002) 

Diffuse interface analysis of crystal nucleation in hard-sphere liquid

László Gránásy and Tamás Pusztai

Research Institute for Solid State Physics and Optics, H 1525 Budapest, POB 49, Hungary

 We show that the increase of the interface free energy with deviation from equilibrium seen in recent Monte Carlo simulations [S. Auer and D. Frenkel, Nature, London, 413, 711 (2001)] can be recovered if the molecular scale diffuseness of the crystal liquid interface is considered. We compare two models, Gránásy’s phenomenological diffuse interface theory, and a density functional theory that relies on the type of Ginzburg-Landau expansion for fcc nucleation, that Shih et al. introduced for bcc crystal. It is shown that, in the range of Monte Carlo simulations, the nucleation rate of the stable fcc phase is by several orders of magnitude higher than for the metastable bcc phase, seen to nucleate first in other fcc systems. The nucleation barrier that the diffuse interface theories predict for small deviations from equilibrium is in far better agreement with the simulations than the classical droplet model. The behavior expected at high densities is model dependent. Gránásy s phenomenological diffuse interface theory indicates a spinodal point close to glass transition, while a nonsingular behavior is predicted by the density functional theory with constant Ginzburg-Landau coefficients. Remarkably, a minimum of the nucleation barrier, similar to the one seen in polydisperse systems, occurs if the known density dependence of the Ginzburg-Landau coefficients is considered. [J. Chem. Phys. B, 117, 11121, (2002)].

Phase Field Theory of Nucleation and Growth in Binary Alloys

László Gránásy,¹ Tamás Börzsönyi,^1,2 and Tamás Pusztai¹

¹Research Institute for Solid State Physics and Optics, POB 49, H-1525 Budapest, Hungary
²Groupe de Physique des Solides, CNRS UMR 75-88, Universités Paris VI at VII, Tour 23, 2 place Jussieu,75251, Paris Cedex 05, France

 

We present a phase field theory for binary crystal nucleation. Using the physical interface thickness, we achieve quantitative agreement with computer simulations and experiments for unary and binary substances. Large-scale numerical simulations are performed for multi-particle freezing in alloys. We deduce the Kolmogorov exponents for dendritic solidification and for the "soft-impingement" of crystallites interacting via diffusion fields.  [Presented at International Workshop on "Computational Physics of Transport and Interface Dynamics" February18-March 8, 2002. MPIPKS Dresden, Germany; Appeared in Interface and Transport Dynamics, edited by H. Emmerich, B. Nestler and M. Schreckenberg, Lecture Notes in Computational Science and Engineering, 32, Springer, Berlin, (2003) pp 190-195.] 

Growth of “dizzy dendrites” in a random field of foreign particles

László Gránásy,¹ Tamás Pusztai^,1 James A. Warren,² Jack F. Douglas,³ Tamás Börzsönyi,¹ and Vincent Ferreiro⁴

 

¹Research Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary
²Metallurgy and ³Polymers Divisions, National Institute of Standards and Technology, Gaithersburg, Maryland 20899,USA
⁴Laboratoire de Structure et Properiétés de l Etat Solide, CNRS, Batiment C6, 59655 Villeneuve d Ascq, France

 

Microstructure plays an essential role in determining the properties of crystalline materials. A widely used method to influence microstructure is the addition of nucleating agents1. Observations on films formed from clay polymer blends indicate that particulate additives, in addition to serving as nucleating agents, may also perturb crystal growth, leading to the formation of irregular dendritic morphologies. Here we describe the formation of these dizzy dendrites using a phase-field theory, in which randomly distributed foreign particle inclusions perturb the crystallization by deflecting the tips of the growing dendrite arms. This mechanism of crystallization, which is verified experimentally, leads to a polycrystalline structure dependent on particle configuration and orientation. Using computer simulations we demonstrate that additives of controlled crystal orientation should allow for a substantial manipulation of the crystallization morphology. [Nature Materials, 2,  92 (2003)].

Phase field theory of crystal nucleation in hard sphere liquid

László Gránásy,¹ Tamás Pusztai,¹ Gyula Tóth,¹ Zoltán Jurek,¹ Massimo Conti,² and Bjørn Kvamme³

 

¹Research Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary
²Dipartimento di Matematica e Fisica, Universita  di Camerino, and Istituto Nazionale di Fisica della Materia, Via Madonna delle Carceri, I-62032, Camerino, Italy
³University of Bergen, Department of Physics, Allégaten 55, N-5007 Bergen, Norway

The phase field theory of crystal nucleation described in L. Gránásy, T. Börzsönyi, and T. Pusztai, Phys. Rev. Lett. 88, 206105 (2002) is applied for nucleation in hard-sphere liquids. The exact thermodynamics from molecular dynamics is used. The interface thickness for phase field is evaluated from the cross-interfacial variation of the height of the singlet density peaks. The model parameters are fixed in equilibrium so that the free energy and thickness of the (111), (110), and (100) interfaces from molecular dynamics are recovered. The density profiles predicted without adjustable parameters are in a good agreement with the filtered densities from the simulations. Assuming spherical symmetry, we evaluate the height of the nucleation barrier and the Tolman length without adjustable parameters. The barrier heights calculated with the properties of the (111) and (110) interfaces envelope the Monte Carlo results, while those obtained with the average interface properties fall very close to the exact values. In contrast, the classical sharp interface model considerably underestimates the height of the nucleation barrier. We find that the Tolman length is positive for small clusters and decreases with increasing size, a trend consistent with computer simulations. [Journal of Chemical Physics, 119,  10376 (2003)].

Phase-field models for eutectic solidification

Daniel Lewis,¹ Tamás Pusztai,² László Gránásy,² James A. Warren,¹ and William Boettinger¹

 

¹Metallurgy and Polymers Divisions, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

²Research Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary

This article discusses two methods for modeling eutectic solidification using the phase-field approach. First, a multi-phase-field model is used to study the three-dimensional morphological evolution of binary eutectics. Performing the calculations in three dimensions allows observation of both lamellar and rod-like structures as well as transient phenomena such as lamellar fault motion, rod-branching, and nucleation or elimination of phases as solidification progresses. The second approach models multiple eutectic grains where the crystallizing phases have an orientation relationship. This approach is promising for modeling complex solidification microstructures. [JOM, 56, 34-39 (2004)].

 

 

 

 

Kinetics of solid hydrate formation by carbon dioxide: Phase field theory of hydrate nucleation and magnetic resonance imaging

B. Kvamme,¹ A. Graue,¹ E. Aspenes,¹ T. Kuynetsova,¹ L. Gránásy,² G. Tóth,² T. Pusztai,² and G. Tegze²

 

¹University of Bergen, Department of Physics, Allégaten 55, N-5007 Bergen, Norway
²Research Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary

In the course of developing a general kinetic model of hydrate formation/reaction that can be used to establish/ optimize technologies for the exploitation of hydrate reservoirs, two aspects of CO2 hydrate formation have been studied. (i) We developed a phase field theory for describing the nucleation of CO2 hydrate in aqueous solutions. The accuracy of the model has been demonstrated on the hard-sphere model system, for which all information needed to calculate the height of the nucleation barrier is known accurately. It has been shown that the phase field theory is considerably more accurate than the sharp-interface droplet model of the classical nucleation theory. Starting from realistic estimates for the thermodynamic and interfacial properties, we have shown that under typical conditions of CO2 formation, the size of the critical fluctuations (nuclei) is comparable to the interface thickness, implying that the droplet model should be rather inaccurate. Indeed the phase field theory predicts considerably smaller height for the nucleation barrier than the classical approach. (ii) In order to provide accurate transformation rates to test the kinetic model under development, we applied magnetic resonance imaging to monitor hydrate phase transitions in porous media under realistic conditions. The mechanism of natural gas hydrate conversion to CO2-hydrate implies storage potential for CO2 in natural gas hydrate reservoirs, with the additional benefit of methane production. We present the transformation rates for the relevant processes (hydrate formation, dissociation and recovery). [Phys. Chem. Chem. Phys., 6,  2327 (2004)].

A general mechanism for polycrystalline growth

László Gránásy,¹ Tamás Pusztai,¹ Tamás Börzsönyi,¹ James A. Warren² and Jack F. Douglas³

 

¹Research Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary
²Metallurgy and ³Polymers Divisions, National Institute of Standards and Technology, Gaithersburg, Maryland 20899,USA

Most research into microstructure formation during solidification has focused on single-crystal growth ranging from faceted crystals to symmetric dendrites. However, these growth forms can be perturbed by heterogeneities, yielding a rich variety of polycrystalline growth patterns. Phase-field simulations show that the presence of particulates (for example, dirt) or a small rotational translational mobility ratio (characteristic of high supercooling) in crystallizing fluids give rise to similar growth patterns, implying a duality in the growth process in these structurally heterogeneous fluids. Similar crystallization patterns are also found in thin polymer films with particulate additives and pure films with high supercooling. This duality between the static and dynamic heterogeneity explains the ubiquity of polycrystalline growth patterns in polymeric and other complex fluids. [Nature Materials, 3, 645 (2004)].

 

Nucleation and polycrystalline formation in binary phase field theory

László Gránásy,¹ Tamás Pusztai,¹ Tamás Börzsönyi,¹ James A. Warren,² Bjørn Kvamme,³ and P.F. James⁴

¹Research Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary
²Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899,USA
³University of Bergen, Department of Physics, Allégaten 55, N-5007 Bergen, Norway
⁴Glass Research Centre, Department of Engineering Materials, The University of Sheffield, Sir Robert Hadfield Building, Mappin Street, Sheffield S1 3JD, UK

We present a phase field theory for the nucleation and growth of one and two phase crystals solidifying with different crystallographic orientations in binary alloys. The accuracy of the model is tested for crystal nucleation in single component systems. It is shown that without adjustable parameters the height of the nucleation barrier is predicted with reasonable accuracy. The kinetics of primary solidification is investigated as a function of model parameters under equiaxial conditions. Finally, we study the formation of polycrystalline growth morphologies (disordered dendrites, spherulites and fractal-like aggregates). [Phys. Chem. Glass, 45,   107-115 (2004)].

 

We thank V. Ferreiro and J. F. Douglas for the experimental images (darker pictures).

 

 

 

 

Modelling polycrystalline solidification using phase field theory

László Gránásy,¹ Tamás Pusztai,¹ and James A. Warren,²

¹Research Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary
²Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899,USA

We review recent advances made in the phase field modelling of polycrystalline solidification. Areas covered include the development of theory from early approaches that allow for only a few crystal orientations, to the latest models relying on a continuous orientation field and a free energy functional that is invariant to the rotation of the laboratory frame. We discuss a variety of phenomena, including homogeneous nucleation and competitive growth of crystalline particles having different crystal orientations, the kinetics of crystallization, grain boundary dynamics, and the formation of complex polycrystalline growth morphologies including disordered (dizzy) dendrites, spherulites, fractal-like polycrystalline aggregates, etc. Finally, we extend the approach by incorporating walls, and explore phenomena such as heterogeneous nucleation, particle front interaction, and solidification in confined geometries (in channels or porous media). [J.Phys. Condens. Matter 16, R1205 (2004)]

Multiphase solidification in multicomponent alloys

U. Hecht,¹ L. Gránásy,² T. Pusztai,² B. Böttger,¹ M. Apel,¹ V. Witusiewicz,¹ L. Ratke,³ J. De Wilde,⁴ L. Froyen,⁴ D. Camel,⁵ B. Drevet,⁵ G. Faivre,⁶ S.G. Fries,¹ B. Legendre,⁷ and S. Rex¹

¹ACCESS e.V, Aachen, Germany
²Research Institute for Solid State Physics and Optics of the Hungarian Academy of Sciences, Budapest, Hungary
³Institute of Space Simulation DLR Köln, Germany
⁴Departement MTM, Katholieke Universiteit Leuven, Faculteit Toegepaste Wetenschappen, Leuven, Belgium
⁵CEA-Grenoble, Grenoble, France
⁶Groupe de Physique des Solides (GPS), Université Paris 6, Paris, France
⁷Laboratoire de Chimie Physique Minérale et Bioinorganique, EA 401, Faculté de Pharmacie, Chatenay-Malabry, France

Multiphase solidification in multicomponent alloys is pertinent to many commercial materials and industrial processes, while also raising challenging questions from a fundamental point of view. Within the past few years, research activities dedicated to multiphase solidification of ternary and multicomponent alloys experienced considerable amplification. This paper gives an overview of our present understanding in this field and the experimental techniques and theoretical methods research relies on. We start with an introduction to thermodynamic databases and computations and emphasize the importance of thermophysical property data. Then, we address pattern formation during coupled growth in ternary alloys and cover microstructure evolution during successive steps of phase formation in solidifying multicomponent alloys. Subsequently, we review advances made in phase field modeling of multiphase solidification in binary and multicomponent alloys, including various approaches to crystal nucleation and growth. Concluding, we address open questions and outline future prospects on the basis of a close interaction among scientists investigating the thermodynamic, thermophysical and microstructural properties of these alloys. [Materials Science and Engineering R 46, 1 (2004)]

Nucleation and the solid-liquid free energy

 

David T. Wu,¹ László Gránásy,² and Frans Spaepen³

 

This article reviews the current understanding of the fundamentals of nucleation theory and its use to extract values for the solid liquid interfacial free energy from experimental and simulation data. [MRS Bulletin, December 2004]

 

 

 

 

 

 

 

 

 

Growth and form of spherulites

 

L. Gránásy,¹ T. Pusztai,¹ G. Tegze,¹  J.A. Warren² and J.F. Douglas³

 

¹Research Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary
²Metallurgy and ³Polymers Divisions, National Institute of Standards and Technology, Gaithersburg, Maryland 20899,USA

 

Many structural materials (metal alloys, polymers, minerals, etc.) are formed by quenching liquids into crystalline solids. This highly nonequilibrium process often leads to polycrystalline growth patterns that are broadly termed "spherulites" because of their large-scale average spherical shape. Despite the prevalence and practical importance of spherulite formation, only rather qualitative concepts of this phenomenon exist. It is established that phase field methods naturally account for diffusional instabilities that are responsible for dendritic single-crystal growth. However, a generalization of this model is required to describe spherulitic growth patterns, and in the present paper we propose a minimal model of this fundamental crystal growth process. Our calculations indicate that the diversity of spherulitic growth morphologies arises from a competition between the ordering effect of discrete local crystallographic symmetries and the randomization of the local crystallographic orientation that accompanies crystal grain nucleation at the growth front  growth front nucleation  (GFN). This randomization in the orientation accounts for the isotropy of spherulitic growth at large length scales and long times. In practice, many mechanisms can give rise to GFN, and the present work describes and explores three physically prevalent sources of disorder that lead to this kind of growth. While previous phase field modeling elucidated two of these mechanisms - disorder created by particulate impurities or other static disorder or by the dynamic heterogeneities that spontaneously form in supercooled liquids  (even pure ones) -  the present paper considers an additional mechanism, crystalline branching induced by a misorientation-dependent grain boundary energy, which can significantly affect spherulite morphology. We find the entire range of observed spherulite morphologies can be reproduced by this generalized phase field model of polycrystalline growth. [Phys. Rev. E  72, 011605 (2005)]

Phase field theory of polycrystalline solidification in three dimensions

T. Pusztai, G. Bortel and L. Gránásy

Research Institute for Solid State Physics and Optics H-1525 Budapest, POB 49, Hungary

A phase field theory of polycrystalline solidification is presented that describes the nucleation and growth of anisotropic particles with different crystallographic orientation in 3D dimensions. As opposed with the two-dimensional case, where a single orientation field suffices, in three dimensions, minimum three fields are needed. The free energy of grain boundaries is assumed to be proportional to the angular difference between the adjacent crystals expressed here in terms of the differences of the four symmetric Euler parameters. The equations of motion for these fields are obtained from variational principles. Illustrative calculations are performed for polycrystalline solidification with dendritic, needle and spherulitic growth morphologies. [Europhys. Lett. 71, 131 (2005)]

 

 

Phase field modeling of polycrystalline freezing

 

T. Pusztai, G. Bortel and L. Gránásy

 

Research Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary

 

The formation of two and three-dimensional polycrystalline structures are addressed within the framework of the phase field theory. While in two dimensions a single orientation angle suffices to describe crystallographic orientation in the laboratory frame, in three dimensions, we use the four symmetric Euler parameters to define crystallographic orientation. Illustrative simulations are performed for various polycrystalline structures including simultaneous growth of randomly oriented dendritic particles, the formation of spherulites and crystal sheaves. [Materials Science and Engineering A 413–414, 412–417 (2005)]

Phase field simulation of liquid phase separation with fluid flow

 

G. Tegze, T. Pusztai and L. Gránásy

 

Research Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary

 

A phase-field theory of binary liquid phase separation coupled to fluid flow is presented. The respective Cahn–Hilliard-type and Navier–Stokes equations are solved numerically. We incorporate composition and temperature dependent capillary forces. The free energies of the bulk liquid phases are taken from the regular solution model. In the simulations, we observe Marangoni motion, and direct and indirect hydrodynamic interactions between the droplets. We find that coagulation is dramatically accelerated by flow effects. Possible extension of the model to solidification is discussed.  [Materials Science and Engineering A 413–414, 418–422 (2005)]

 

 

 

 

 

Phase field theory of crystal nucleation and polycrystalline growth: A review

L. Gránásy,¹ T. Pusztai,¹ T. Börzsönyi,¹ G. Tóth,¹ G. Tegze,¹ J.A. Warren,² and J.F. Douglas²

¹Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

²National Institute of Standards and Technology, Gaithersburg, Maryland 20899

We briefly 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. [J. Mater. Res.,  21, 309 (2006)]

 

 

 

Multiscale approach to CO2 hydrate formation in aqueous solution: Phase field theory and molecular dynamics. Nucleation and growth

György Tegze,¹ Tamás Pusztai,¹ Gyula Tóth,¹ László Gránásy,¹ Atle Svandal,² Trygve Buanes,² Tatyana Kuznetsova,² and Bjørn Kvamme²

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

A phase field theory with model parameters evaluated from atomistic simulations/experiments is applied to predict the nucleation and growth rates of solid CO2 hydrate in aqueous solutions under conditions typical to underwater natural gas hydrate reservoirs. It is shown that under practical conditions a homogeneous nucleation of the hydrate phase can be ruled out. The growth rate of CO2 hydrate dendrites has been determined from phase field simulations as a function of composition while using a physical interface thickness 0.85±0.07 nm evaluated from molecular dynamics    simulations. The growth rate extrapolated to realistic supersaturations is about three orders of magnitude larger than the respective experimental observation. A possible origin of the discrepancy is discussed. It is suggested that a kinetic barrier reflecting the difficulties in building the complex crystal structure is the most probable source of the deviations. [J. Chem. Phys. 124, 234710 (2006)]

Phase field theory of polycrystalline freezing in three dimensions

 

Tamás Pusztai, Gábor Bortel and László Gránásy

Research Institute for Solid State Physics and Optics; H-1525 Budapest, POB 49, Hungary

A phase field theory, we proposed recently to describe nucleation and growth in three dimensions (3D), has been used to study the formation of polycrystalline patterns in the alloy systems Al-Ti and Cu-Ni. In our model, the free energy of grain boundaries is assumed proportional to the angular difference between the adjacent crystals expressed in terms of the differences of the four symmetric Euler parameters called quaternions. The equations of motion for these fields have been obtained from variational principles. In the simulations cubic crystal symmetries are considered. We investigate the evolution of polydendritic morphology, present simulated analogies of the metallographic images, and explore the possibility of modeling solidification in thin layers. Transformation kinetics in the bulk and in thin films is discussed in terms of the Johnson-Mehl-Avrami-Kolmogorov approach. [Modeling of Casting, Welding and Advanced Solidification Processes- XI, TMS 409 (2006)]

Phase field theory of liquid phase separation and solidification with melt flow

György Tegze and  László Gránásy

Research Institute for Solid State Physics and Optics; H-1525 Budapest, POB 49, Hungary

A phase-field theory of binary liquid phase separation and solidification coupled to fluid flow is presented. The respective equations of motion and Navier-Stokes equations are solved numerically. We incorporate composition and temperature dependent capillary forces. The free energies of the bulk liquid phases are taken from the regular solution model. In the simulations, we observe Marangoni motion of the droplets, and direct and indirect hydrodynamic interactions between the droplets. We observe that capillary effects dramatically accelerate droplet coagulation and that solidification interacts with liquid phase separation.  [Modeling of Casting, Welding and Advanced Solidification Processes- XI, TMS 513 (2006)]

Polycrystalline patterns in far-from-equilibrium freezing: a phase field study

L. Gránásy,¹ T. Pusztai,¹ T. Börzsönyi,¹ G. Tóth,¹ G. Tegze,¹ J.A. Warren,² and J.F. Douglas^2
 
1Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
²National Institute of Standards and Technology, Gaithersburg, Maryland 20899

 

We discuss the formation of polycrystalline microstructures within the framework of phase field theory. First, the model is tested for crystal nucleation in a hard sphere system. It is shown that, when evaluating the model parameters from molecular dynamics simulations, the phase field theory predicts the nucleation barrier for hard spheres accurately. The formation of spherulites is described by an extension of the model that incorporates branching with a definite orientational mismatch. This effect is induced by a metastable minimum in the orientational free energy. Spherulites are an extreme example of polycrystalline growth, a phenomenon that results from the quenching of orientational defects (grain boundaries) into the solid as the ratio of the rotational to the translational diffusion coefficient is reduced, as is found at high undercoolings. It is demonstrated that a broad variety of spherulitic patterns can be recovered by changing only a few model parameters. [Philos. Mag. 86 3757 (2006)]

Phase field theory of nucleation and polycrystalline pattern formation

L. Gránásy, T. Pusztai and T. Börzsönyi

Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

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, Edited by Michael Rieth and Wolfram Schommers American Scientific Publishers, Stevenson Ranch, CAL, 2006, Volume 9: Pages (525-572)]

Phase field theory of heterogeneous crystal nucleation

L. Gránásy,¹ T. Pusztai,¹ D. Saylor,² and J.A. Warren<sup>3

1</sup>Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

²Food and Drug Administration, Rockville, Maryland 20852, USA
³National 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. [Phys. Rev. Lett. 98, 035703 (2007) ]

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

Gy. I. Tóth and L. Gránásy,

Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

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. [J. Chem. Phys. 127, 074709 (2007)]

Phase field theory of interfaces and crystal nucleation in a eutectic system of fcc structure: II. Nucleation in the metastable liquid immiscibility region

G. I. Tóth and L. Gránásy,

Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

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. [J. Chem. Phys. 127, 074710 (2007)]

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

T. Pusztai,¹ G. Tegze,² G. I. Tóth,¹ L. Környei,¹ G. Bansel,² Z. Fan,² and L. Gránásy^2
 
1Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary;
²Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK

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. [J. Phys.: Condens. Matter 20, 404205 (2008)]

 

 

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

G. Tegze,¹ G. Bansel,¹ G. I. Tóth,² T. Pusztai,² Z. Fan,¹ and L. Gránásy¹

¹Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK
²Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

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. [J. Comput. Phys. 228, 1612 (2009)]

Phase field approach to heterogeneous nucleation in alloys.

J. A. Warren,¹ T. Pusztai,² L. Környei,² and L. Gránásy^3
 
1Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
²Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
³ Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK

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. [Phys. Rev. B 79, 014204 (2009)]

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

G. I. Tóth¹ and L. Gránásy²

¹Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
²Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK

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/s² to 0.56 kT/s² [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. [J. Phys. Chem. B 113, 5141 (2009)]

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

G. Tegze,¹ L. Gránásy,¹ G. I. Tóth,² F. Podmaniczky,² A. Jaatinen,³ T. Ala-Nissila,³  and T. Pusztai<sup>2

1</sup>Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK
²Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
³Department 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. [Phys. Rev. Lett. 103, 035702 (2009)]

 

Classical density functional theory methods in soft and hard matter

M. Haataja,¹ L. Gránásy,^2,3 and H. Löwen<sup>4

1</sup>Department 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
²Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
³BCAST, Brunel University, Uxbridge UB8 3PH, UK
⁴Department of Theoretical Physics, 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. [ J. Phys.: Condens. Matter 22, 360301 (2010)]

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

G. I. Tóth,¹ G. Tegze,¹ T. Pusztai,¹ G. Tóth,²  and L. Gránásy^1,3

¹Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
²Institute of Chemistry, Eötvös University, PO Box 32, H-1518 Budapest, Hungary

³Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK

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. [ J. Phys.: Condens. Matter 22, 364101 (2010).]

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

L. Gránásy,^1,2 G. Tegze,¹ G. I. Tóth,¹ and T. Pusztai¹

¹Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
²Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK

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. [ Philos. Mag. 91, 123-149 (2011).]

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

G. Tegze,¹ L. Gránásy,^1,2 G. I. Tóth,¹ J. F. Douglas,³ and T. Pusztai¹

¹Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
²Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK

³Polymers 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. [ Soft Matter 7, 1789-1799 (2011).]

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

G. I. Tóth,¹ J. R. Morris,² and L. Gránásy^1,3

¹Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

²Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA
³BCAST, 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. [ Phys. Rev. Lett. 105, 045701 (2011).]