Numerical research of solidification dynamics with anisotropy and thermal fluctuations (original) (raw)

Phase-Field Simulation of Solidification With Density Change

Heat Transfer, Volume 3, 2004

Phase-field models of solidification with convection often assume the existence of a single (mixture) velocity at any location inside the diffuse interface, and the phase-field, φ, is advected by this mixture velocity. In this paper, the advection of the phase-field is examined for a one-dimensional normal flow to a solidification front induced by a density difference between the solid and liquid. It is found that the results from a phase-field model that assumes a single velocity inside the diffuse interface are generally not in agreement with the sharp interface condition for the kinetic undercooling of the front in the presence of unequal densities, regardless of the interface width. By introducing a two-phase approach, where the solid and liquid are assumed to coexist inside the diffuse interface with different velocities, good agreement with the sharp interface condition is obtained irrespective of the density ratio between the two phases.

Phase field modeling of polycrystalline freezing

Materials Science and Engineering: A, 2005

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.

Derivation of the phase-field-crystal model for colloidal solidification

Physical Review E, 2009

The phase-field-crystal model is by now widely used in order to predict crystal nucleation and growth. For colloidal solidification with completely overdamped individual particle motion, we show that the phase-fieldcrystal dynamics can be derived from the microscopic Smoluchowski equation via dynamical densityfunctional theory. The different underlying approximations are discussed. In particular, a variant of the phasefield-crystal model is proposed which involves less approximations than the standard phase-field-crystal model. We finally test the validity of these phase-field-crystal models against dynamical density-functional theory. In particular, the velocities of a linear crystal front from the undercooled melt are compared as a function of the undercooling for a two-dimensional colloidal suspension of parallel dipoles. Good agreement is only obtained by a drastic scaling of the free energies in the phase-field-crystal model in order to match the bulk freezing transition point.

A grand potential approach to phase-field modeling of rapid solidification

Journal of Non-Equilibrium Thermodynamics, 2014

Rapid solidi cation occurs under large driving force of transformation from the metastable undercooled liquid phase to the stable crystalline state. Using a formalism of extended irreversible thermodynamics, a phase-eld model of rapid solidi cation in binary systems is derived. An entropy approach together with a grand potential density of a binary system is used to obtain the main governing equations of the model. Special attention is paid to equations of a rapidly solidifying binary system which are accompanied by essential deviations from local equilibrium in the transport of the conservative variables (such as inner energy and mass) and in the dynamics of non-conservative variables (such as phase eld). The obtained equations are analyzed and compared with recent models and outcomes based on the grand potential approach to solidi cation.

Phase-Field Modeling of Solidification under Stress | NIST

Acta Materialia, 2006

We consider a phase field model that includes a stress field during nonisothermal phase transformation of a single-component system. The model is applied to the solidification and melting of confined spherical volumes, where sharp interface solutions can be obtained and compared with the results of the phase field simulations. Numerical solutions to the phase field model for a spherically symmetric geometry have been obtained, with particular emphasis on the computation of surface energy, surface stress, and surface strain. The analysis of the equilibrium states for the phase field model allows us to obtain the value of the surface energy in the presence of stress, which can then be compared to the analogous calculation of the energy of a planar liquid-solid interface. It is also demonstrated that modeling the liquid as a solid with zero shear modulus is realistic by comparing the long-range stress fields in phase field calculations with those calculated using sharp interface models of either a coherent or a relaxed liquid-solid interface.

Three-dimensional phase-field simulations of directional solidification

2004

The phase-field method has become in recent years the method of choice for simulating microstructural pattern formation during solidification. One of its main advantages is that time-dependent three-dimensional simulations become feasible. This makes it possible to address long-standing questions of pattern stability. Here, we investigate the stability of hexagonal cells and eutectic lamellae. For cells, it is shown that the

Solid/Liquid Phase Change: Recent Studies and Models

2007

Some problems related to solid/liquid phase change are presented. Attention is focused on interface modeling for numerical analysis and one-dimensional directional growing and melting. Microgravity relevance of some situations is emphasized. It is shown, in particular, that in some circumstances melting is not the simple reversal of crystal growth due to some (still poorly known) phenomena (nucleation and growth of liquid droplets in the bulk, solid and liquid dendrites due to a morphological instability of the phase boundary). Relevant mathematical models are discussed and described (to a certain extent) for analysis and/or characterization of these phenomena when they are disjoint or partially combined. Some effort is provided to model phenomena occurring at both the microscopic and macroscopic scale lengths.

Phase-field crystal modeling and classical density functional theory of freezing

Physical Review B, 2007

In this paper the relationship between the classical density functional theory of freezing and phase field modeling is examined. More specifically a connection is made between the correlation functions that enter density functional theory and the free energy functionals used in phase field crystal modeling and standard models of binary alloys (i.e., regular solution model). To demonstrate the properties of the phase field crystal formalism a simple model of binary alloy crystallization is derived and shown to simultaneously model solidification, phase segregation, grain growth, elastic and plastic deformations in anisotropic systems with multiple crystal orientations on diffusive time scales.

Numerical investigations of planar solidification of an undercooled liquid

Space technology and applications international forum (STAIF - 97), 1997

We investigate evolution of a planar interface during unstable solidification of a pure undercooled liquid between two parallel plates. The governing equations are solved using a front tracking/finite difference technique that allows discontinuous material properties between the phases and interfacial anisotropy. The simulations produce some of the futures of the dendritic solidification which are in good qualitative agreement with the works of the previous investigators. The effects of the physical parameters on the crystal growth and interface instability are also examined.

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

Philosophical Magazine, 2006

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.