Narrow Results

A phase field model is used to study dendritic growth in a media with impurities. The model consists of a square lattice where a parameter $\psi$ can take values between $0$ and $1$ at each site. A site is in the solid phase for $\psi >1$, in the liquid phase for $\psi =0$, and the solid-liquid interface is expressed by $0<\psi <1$. A fraction of the sites are considered impurities that cannot be solidified, i.e.$\psi$ is fixed and taken as zero. These impurities are distributed randomly. As the probability $p$ of impure sites in the lattice increases, the growth loses its dendritic characteristic. It is shown that the perimeter of the growing solid goes from quadratic to a linear function with time. It was also found that as the probability of impurities reaches $p=0.004$, the solid undergoes a transition from anisotropic to isotropic growth.

Interfacial pattern formation in phase transition and crystal growth and material science is one of the most important subjects in the broad field of nonlinear science. This subject involves the concepts and issues, include the basic states, global stability, limiting-state selection, quantization conditions of eigenvalues, scaling law and free boundary problems of dynamic system far away from the equilibrium state.

This talk attempts to explore these issues through the two prototype problems: (1). dendritic growth from melt; (2). disc-like crystal growth from melt. These problems are highly challenging fundamental problems in condensed matter physics and material science, which have preoccupied many investigators from various areas of science, including applied mathematics for a long period of time.

We shall summarize the major results achieved in terms of a unified systematic asymptotic approach during the last decade. For the case of dendritic growth, these results described the wave-characteristics of interface evolution and led to the so-called interfacial wave (IFW) theory.

The final solidification structures of Vacuum Arc Remelting (VAR) ingots depend on the temperature distribution and fluid motion within the molten pool. In this paper, a three-dimensional multi-physics macroscale model for VAR is developed, based on the modular CFD software PHYSICA. This model is used to provide estimates of process parameters and to study complex physical phenomena, such as liquid metal flow with turbulence, heat transfer, solidification, and magnetohydrodynamics in the VAR process. The macromodel is coupled to a microscale solidification model. The micromodel combines stochastic nucleation and a modified decentred square/octahedron method to describe dendritic growth with a finite difference computation of solute diffusion. The resulting multiscale model allows prediction of the formation of microstructures in the solidifying mushy zone. This gives a better understanding of the whole VAR process from operational conditions to final ingot microstructures, as well as an essential first step in defect prediction.

A two dimensional (2D) cellular automaton (CA) - lattice Boltzmann (LB) model is presented to investigate the effects of forced melt convection on the solutal dendritic growth. In the model, the CA approach of simulating the dendritic growth is incorporated with the kinetic-based lattice Boltzmann method (LBM) for numerically solving the melt flow and solute transport. Two sets of distribution functions are used in the LBM to model the convective-diffusion phenomena during dendritic growth. After validating the model by comparing the numerical results with the theoretical solutions, it is applied to simulate the single and multi dendritic growth of Al-Cu alloys without and with a forced convection. The typical asymmetric growth features of convective dendrite are reproduced and the dendritic morphology is strongly influenced by melt convection. The simulated convective multi dendritic features by the present model are also compared with that by the CA-NS model. The present model is found to be more computationally efficient and numerically stable than the CA-NS model.

The non-isothermal polycrystalline solidification of a binary alloy is simulated by employing a phase field model which takes into account the heat transition and the random crystallographic orientation. The stochastic nucleation is taken into account in the simulation through the Poisson seeding algorithm and a kinetic calculation for binary melts based on the classical nucleation theory. Different microstructures are obtained under various cooling conditions. It is found that the grain structure becomes finer with increasing the cooling rate, which agrees with experimental result.

On the basis of Xu’s interfacial wave theory, the stability of dendritic growth in a convective binary alloy melt with buoyancy effect is studied using the asymptotic method. The resulting asymptotic solution of equations reveals that the stability mechanism of dendritic growth in the binary alloy melt with buoyancy-driven convection is similar to that in a pure melt. Dendritic growth is stable above and unstable below a critical stability number ${𝜀}_{\ast }$, which is determined by the quantization condition. In particular, there is a critical morphological number in the binary alloy melt. When the morphological number is less than the critical morphological number, the tip growth velocity increases, the tip curvature radius and oscillation frequency decrease, and the interface becomes thinner and smooth. When the morphological number is larger than the critical morphological number, the tip growth velocity decreases, the tip curvature radius and oscillation frequency increase, and the interface becomes fatter and rough. The result demonstrates that in a microgravity environment, there is a critical initial concentration such that below it thermal diffusion dominates, the tip growth velocity increases, the tip curvature radius and oscillation frequency decrease, and the interface becomes thinner and smooth; above it, solute diffusion dominates, the tip growth velocity decreases, the tip curvature radius and oscillation frequency increase, and the interface becomes fatter and rough.

A numerical method is presented to simulate dendritic crystal growth on a fixed mesh. The hyperbolic convective equation for the continuous phase field which identifies the solid and liquid phases is solved by the least-squares finite element method, and the heat conduction equation for the temperature field is solved by the Galerkin finite element method. Without special treatments this method can handle complicated interfacial shapes and physical features.