Bálint Korbuly¹, Tamás Pusztai¹, Gyula Tóth², Hervé Henry³, Mathis Plapp³, László Gránásy^1,4

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.

Frigyes Podmaniczky¹, Gyula Tóth², Tamás Pusztai¹, László Gránásy^1,3

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.

László Gránásy^1,2, 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.

Gyula Tóth¹, Tamás Pusztai², László Gránásy^2,3

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.