Phase-field modeling of eutectic structures on the nanoscale: the effect of anisotropy

László Rátkai1, Gyula Tóth2, László Környei3, Tamás Pusztai1, 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

A simple phase-field model is used to address anisotropic eutectic freezing on the nanoscale in two (2D) and three dimensions (3D). Comparing parameter-free simulations with experiments, it is demonstrated that the employed model can be made quantitative for Ag-Cu. Next, we explore the effect of material properties, and the conditions of freezing on the eutectic pattern. We find that the anisotropies of kinetic coefficient and the interfacial free energies (solid-liquid and solid-solid), the crystal misorientation relative to pulling, the lateral temperature gradient, play essential roles in determining the eutectic pattern. Finally, we explore eutectic morphologies, which form when one of the solid phases are faceted, and investigate cases, in which the kinetic anisotropy for the two solid phases are drastically different.

Investigating Nucleation Using the Phase-Field Method

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 first order phase transitions, like freezing of liquids, melting of solids, phase separation in alloys, vapor condensation, etc., start with nucleation, a process in which internal fluctuations of the parent phase lead to formation of small seeds of the new phase. Owing to different size dependence of (negative) volumetric and (positive) interfacial contributions to work of formation of such seeds, there is a critical size, at which the work of formation shows a maximum. Seeds that are smaller than the critical one decay with a high probability, while the larger ones have a good chance to grow further and reach a macroscopic size. Putting it in another way, to form the bulk new phase, the system needs to pass a thermodynamic barrier via thermal fluctuations. When the fluctuations of the parent phase alone lead to transition, the process is called homogeneous nucleation. Such a homogeneous process is, however, scarcely seen and requires very specific conditions in nature or in the laboratory. Usually, the parent phase resides in a container and/or it incorporates floating heterogeneities (solid particles, droplets, etc.). The respective foreign surfaces lead to ordering of the adjacent liquid layers, which in turn may assist the formation of the seeds, a process termed heterogeneous nucleation. Herein, we review how the phase-field techniques contributed to the understanding of various aspects of crystal nucleation in undercooled melts, and its role in microstructure evolution. We recall results achieved using both conventional phase-field techniques that rely on spatially averaged (coarse grained) order parameters in capturing the phase transition, as well as molecular scale phase-field approaches that employ time averaged fields, as happens in the classical density functional theories, including the recently developed phase-field crystal models.

Topics: Heterogeneous nucleation

A Physically Consistent Multiphase-Field Theory of First Order Phase Transitions

Gyula Tóth1, Tamás Pusztai2, Bjørn Kvamme3, László Gránásy2,4

1Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
2Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
3Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
4BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

A multiphase-field theory is presented that describes interface driven multi-domain dynamics. The free energy functional and the dynamic equations are constructed on the basis of criteria of mathematical and physical consistency. First, it is demonstrated that the most widely used multiphase theories are physically inconsistent, therefore, a new theory has to be developed. Combining elements of the investigated models with a new multivariate generalization of the free energy surface results in a general multiphase / multi-component theory, which keeps the variational formalism, reduces / extends naturally on the level of both the free energy functional and the dynamic equations, utilizes arbitrary pairwise equilibrium interfacial properties, features equilibrium = stationary equivalency, and avoids the appearance of spurious phases.

Phase-field crystal modeling of nucleation including homogeneous and heterogeneous processes, and growth front nucleation

László Gránásy1,2, Frigyes Podmaniczky1, 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.

Structural aspects of crystal nucleation in undercooled liquids are explored using a nonlinear hydrodynamic theory of freezing proposed recently, which is based on combining fluctuating hydrodynamics with the phase-field crystal (PFC) theory. It will be shown that unlike the usual PFC models of diffusive dynamics, within the hydrodynamic approach not only the homogeneous and heterogeneous nucleation processes are accessible, but also growth front nucleation, which leads to the formation of differently oriented grains at the front in highly undercooled systems. Formation of dislocations at the solid-liquid interface and the interference of density waves ahead of the crystallization front are responsible for the appearance of new orientations.

Topics: Phase field crystal

Orientation-field models for polycrystalline solidification: grain coarsening and complex growth forms

Bálint Korbuly1, Tamás Pusztai1, Gyula Tóth2, Hervé Henry3, Mathis Plapp3, 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.
3Laboratoire Physique de la Matière Condensée, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France
4BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

We compare two versions of the phase-field theory for polycrystalline solidification, both relying on the concept of orientation fields: one by Kobayashi et al. [Physica D 140 (2000) 141] and the other by Henry et al. [Phys. Rev. B 86 (2012) 054117]. Setting the model parameters so that the grain boundary energies and the time scale of grain growth are comparable in the two models, we first study the grain coarsening process including the limiting grain size distribution, and compare the results to those from experiments on thin films, to the models of Hillert, and Mullins, and to predictions by multiphase-field theories. Next, following earlier work by Gránásy et al. [Phys. Rev. Lett. 88 (2002) 206105; Phys. Rev. E 72 (2005) 011605], we extend the orientation field to the liquid state, where the orientation field is made to fluctuate in time and space, and employ the model for describing of multi-dendritic solidification, and polycrystalline growth, including the formation of “dizzy” dendrites disordered via the interaction with foreign particles.

Topics: Orientation field models, Polycrystalline solidification

Phase-field crystal modeling of heteroepitaxy and exotic modes of crystal nucleation

Frigyes Podmaniczky1, Gyula Tóth2, György Tegze1, 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

We review recent advances made in modeling heteroepitaxy, two-step nucleation, and nucleation at the growth front within the framework of a simple dynamical density functional theory, the Phase-Field Crystal (PFC) model. The crystalline substrate is represented by spatially confined periodic potentials. We investigate the misfit dependence of the critical thickness in the StranskiKrastanov growth mode in isothermal studies. Apparently, the simulation results for stress release via the misfit dislocations fit better to the PeopleBean model than to the one by Matthews and Blakeslee. Next, we investigate structural aspects of two-step crystal nucleation at high undercoolings, where an amorphous precursor forms in the first stage. Finally, we present results for the formation of new grains at the solid-liquid interface at high supersaturations/supercoolings, a phenomenon termed Growth Front Nucleation (GFN). Results obtained with diffusive dynamics (applicable to colloids) and with a hydrodynamic extension of the PFC theory (HPFC, developed for simple liquids) will be compared. The HPFC simulations indicate two possible mechanisms for GFN.

Topics: Phase field crystal

Consistent multiphase-field theory for interface driven multidomain dynamics

Gyula Tóth1, 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

We present a new multiphase-field theory for describing pattern formation in multi-domain and/or multi-component systems. The construction of the free energy functional and the dynamic equations is based on criteria that ensure mathematical and physical consistency. We first analyze previous multiphase-field theories, and identify their advantageous and disadvantageous features. On the basis of this analysis, we introduce a new way of constructing the free energy surface, and derive a generalized multiphase description for arbitrary number of phases (or domains). The presented approach retains the variational formalism; reduces (or extends) naturally to lower (or higher) number of fields on the level of both the free energy functional and the dynamic equations; enables the use of arbitrary pairwise equilibrium interfacial properties; penalizes multiple junctions increasingly with the number of phases; ensures non-negative entropy production, and the convergence of the dynamic solutions to the equilibrium solutions; and avoids the appearance of spurious phases on binary interfaces. The new approach is tested for multi-component phase separation and grain coarsening.

Topics: Emulsion

Phase-field theory of multicomponent incompressible Cahn-Hilliard liquids

Gyula Tóth1, Mojdeh Zarifi, 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, a generalization of the Cahn-Hilliard theory of binary liquids is presented for multicomponent incompressible liquid mixtures. First, a thermodynamically consistent convection-diffusion-type dynamics is derived on the basis of the Lagrange multiplier formalism. Next, a generalization of the binary Cahn-Hilliard free-energy functional is presented for an arbitrary number of components, offering the utilization of independent pairwise equilibrium interfacial properties. We show that the equilibrium two-component interfaces minimize the functional, and we demonstrate that the energy penalization for multicomponent states increases strictly monotonously as a function of the number of components being present. We validate the model via equilibrium contact angle calculations in ternary and quaternary (four-component) systems. Simulations addressing liquid-flow-assisted spinodal decomposition in these systems are also presented.

Topics: Emulsion

Growth control of peptide-nanotube spherulitic films: Experiments and simulations

Netta Hendler1,2, Elad Mentovich1,2, Bálint Korbuly3, Tamás Pusztai3, László Gránásy3,4, Shachar Richter1,2

1Department of Materials Science and Engineering, Faculty of Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
2Center for Nanoscience and Nanotechnology, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
3Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
4BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

Multi-hierarchical self-assembly (MHSA) is a key process responsible for the spontaneous formation of many complex structures. However, because of the complexity of the process, the underlying mechanism remains largely unclear. Thus, a deeper understanding of MHSA is required, especially for the preparation of MHSA systems via bottom-up methodologies. We show here, experimentally and theoretically, that the complex-formation MHSA of peptide nanotube films can be controlled solely by manipulating the experimental parameter of humidity. Furthermore, we identify growth-front nucleation (GFN; the formation of new grains at the perimeter) as the physical background for the observed morphological transitions by correlating experimental observations with phase-field modeling of the morphological evolution. Our findings indicate a simple way to control multi-hierarchical morphologies, crucial for the employment of bottom-up techniques in constructing complex structures for practical applications.

Recent Developments in Modeling Heteroepitaxy/Heterogeneous Nucleation by Dynamical Density Functional Theory

Frigyes Podmaniczky1, Gyula Tóth2, György Tegze1, 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

Crystallization of supersaturated liquids usually starts by epitaxial growth or by heterogeneous nucleation on foreign surfaces. Herein, we review recent advances made in modeling heteroepitaxy and heterogeneous nucleation on flat/modulated surfaces and nanoparticles within the framework of a simple dynamical density functional theory, known as the phase-field crystal model. It will be shown that the contact angle and the nucleation barrier are nonmonotonous functions of the lattice mismatch between the substrate and the crystalline phase. In continuous cooling studies for substrates with lattice mismatch, we recover qualitatively the Matthews–Blakeslee mechanism of stress release via the misfit dislocations. The simulations performed for particle-induced freezing will be confronted with recent analytical results, exploring thus the validity range of the latter. It will be demonstrated that 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

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