Phase-field Model For Solidification Research Articles

Interface stabilization and propagation in phase field models of solidification: resolving the issue of large driving forces

Abstract One of the long-standing problems in the phase field, namely, combining the principles of thermodynamics and capillarity with the numerical aspects of interface propagation in simulations, is re-investigated. Numerical schemes are discussed which allow for stable simulations with arbitrary driving forces, considering or excluding capillarity. We re-investigate a classical stabilization scheme that decouples interface stabilization from curvature evaluation, ensuring stable simulations even under large driving forces. A novel mathematical analysis gives a rigorous estimate for the time stepping and a numerical value of the required stabilization strength. The proposed stabilization scheme is benchmarked for 3-dimensional dendritic growth under directional solidification conditions for different solidification speeds.

Fast prediction of phase equilibrium at varying temperatures for use in multi-component phase field models

A new method for temperature-dependent phase equilibrium prediction for use in multi-component phase field models of solidification is proposed. The method consists of two parts. First, the convex hull method is applied to predict the phase equilibrium at a single temperature. Second, a set of linear equations is developed to extend the equilibrium calculation over a range of temperatures. These linear equations are derived as an extension of the equation used for solidification of binary alloys in approximating the equilibrium state of multi-component systems. Phase field simulations of solidification of a Ti-Al-V-Fe alloy are performed to demonstrate the effectiveness of the present approach under isothermal and continuous cooling conditions. The results are compared against Thermo-Calc calculations, and indicate that a high accuracy of equilibrium prediction is achieved at a single and multiple temperatures, thus demonstrating that this approach can be successfully applied to the multi-component phase field models.

Development of an Anti-Trapping Current for Phase Field Models Using Arbitrary CALPHAD Thermodynamics

Unless corrected by so called anti-trapping currents, phase field models of solidification display a dependence upon the diffuse interface width, δ, used in the simulation. This is most commonly manifest as a reduction in solute partitioning, which is both growth velocity and interface width dependent, resulting in a serious impediment to quantitative simulation. However, such anti-trapping currents are often restricted to very simple materials thermodynamics, appropriate only to dilute ideal solutions. Here we propose a form of the anti-trapping current which can be implemented for arbitrary thermodynamics, including both Redlich-Kister solution phases and sub-lattice models for intermetallic growth. The effect of the new anti-trapping current is illustrated with respect to Pb dendrites growing from a Pb-Sn melt containing either 25% or 30% Sn. The new anti-trapping current is shown to render the solutions independent of the diffuse interface width both with regard to solute partitioning and other growth metrics such as solidification velocity and dendrite tip radius.

Phase field modeling of solidification in multi-component alloys with a case study on the Inconel 718 alloy

Abstract

Phase Field Model for Solidification with Boundary Interface Interaction

By incorporation the surface free energy in the free energy functional, a phase field model for solidification with boundary interface intersection is developed. In this model, the bulk equation is appropriately modified to account for the presence of heat diffusion inside the diffuse interface, and a relaxation boundary condition for the phase field variable is introduced to balance the interface energy and boundary surface energy in the multiphase contact region. The asymptotic analysis is applied on the phase field model to yield the free interface problem with dynamic contact point condition.

Bridging the phase-field and phase-field crystal approaches for anisotropic material systems

In this paper the amplitude representation of the anisotropic phase-field crystal (APFC) model recently proposed as a generalized model for isotropic as well as anisotropic crystal lattice systems is developed. The relationship between the amplitude equations and the standard phase-field model for solidification of pure substances with elasticity effects is derived which provide an explicit connection between the phase-field and APFC models. On the one hand we present a computationally more efficient model and highlight its potential as a bridge between the PFC and phase-field models with anisotropic interface energies and kinetics on the other hand.

A thermodynamically consistent numerical method for a phase field model of solidification

A discretization is presented for the initial boundary value problem of solidification as described in the phase-field model developed by Penrose and Fife (1990) [1] and Wang et al. (1993) [2]. These are models that are completely derived from the laws of thermodynamics, and the algorithms that we propose are formulated to strictly preserve them. Hence, the discrete solutions obtained can be understood as discrete dynamical systems satisfying discrete versions of the first and second laws of thermodynamics. The proposed methods are based on a finite element discretization in space and a midpoint-type finite-difference discretization in time. By using so-called discrete gradient operators, the conservation/entropic character of the continuum model is inherited in the numerical solution, as well as its Lyapunov stability in pure solid/liquid equilibria.

Stable phase field approximations of anisotropic solidification

We introduce unconditionally stable finite element approximations for a phase field model for solidification, which take highly anisotropic surface energy and kinetic effects into account. We hence approximate Stefan problems with anisotropic Gibbs--Thomson law with kinetic undercooling, and quasi-static variants thereof. The phase field model is given by {align*} \vartheta\,w_t + \lambda\,\varrho(\varphi)\,\varphi_t & = \nabla \,.\, (b(\varphi)\,\nabla\, w) \,, \cPsi\,\tfrac{a}\alpha\,\varrho(\varphi)\,w & = \epsilon\,\tfrac\rho\alpha\,\mu(\nabla\,\varphi)\,\varphi_t -\epsilon\,\nabla \,.\, A'(\nabla\, \varphi) + \epsilon^{-1}\,\Psi'(\varphi) {align*} subject to initial and boundary conditions for the phase variable $\varphi$ and the temperature approximation $w$. Here $\epsilon > 0$ is the interfacial parameter, $\Psi$ is a double well potential, $\cPsi = \int_{-1}^1 \sqrt{2\,\Psi(s)}\;{\rm d}s$, $\varrho$ is a shape function and $A(\nabla\,\varphi) = \tfrac12\,|\gamma(\nabla\,\varphi)|^2$, where $\gamma$ is the anisotropic density function. Moreover, $\vartheta \geq 0$, $\lambda > 0$, $a > 0$, $\alpha > 0$ and $\rho \geq 0$ are physical parameters from the Stefan problem, while $b$ and $\mu$ are coefficient functions which also relate to the sharp interface problem. On introducing the novel fully practical finite element approximations for the anisotropic phase field model, we prove their stability and demonstrate their applicability with some numerical results.

Stability and bifurcation of nonconstant solutions to a reaction–diffusion system with conservation of mass

We deal with a two-component system of reaction–diffusion equations with conservation of mass in a bounded domain with the Neumann or periodic boundary conditions. This system is proposed as a conceptual model for cell polarity. Since the system has conservation of mass, the steady state problem is reduced to that of a scalar reaction–diffusion equation with a nonlocal term. That is, there is a one-to-one correspondence between an equilibrium solution of the system with a fixed mass and a solution of the scalar equation. In particular, we consider the case when the reaction term is linear in one variable. Then the equations are transformed into the same equations as the phase-field model for solidification. We thereby show that the equations allow a Lyapunov function. Moreover, by investigating the linearized stability of a nonconstant equilibrium solution, we prove that given a nondegenerate stable equilibrium solution of the nonlocal scalar equation, the corresponding equilibrium solution of the system is stable. We also exhibit global bifurcation diagrams for equilibrium solutions to specific model equations by numerics together with a normal form near a bifurcation point.

Remarks on some open problems in phase-field modelling of solidification

Three different topics in phase-field modelling of solidification are discussed, with particular emphasis on the limitations of the currently available modelling approaches. First, thin-interface limits of two-sided phase-field models are examined, and it is shown that the antitrapping current is, in general, not sufficient to remove all thin-interface effects. Second, orientation-field models for polycrystalline solidification are analysed, and it is shown that the standard relaxational equation of motion for the orientation field is incorrect in coherent polycrystalline matter. Third, it is pointed out that the standard procedure to incorporate fluctuations into the phase-field approach cannot be used in a straightforward way for a quantitative description of nucleation.

Simulação de microestruturas de ligas ternarias pelo método do Campo de Fase

Um modelo do Campo de Fase para a solidificação de ligas ternárias Fe-C-P é proposto. Neste trabalho a relação entre as propriedades do material e os parâmetros do modelo é apresentada. No desenvolvimento do trabalho foi aplicado o modelo para a solidificação de ligas ternárias para a análise da difusividade dos solutos. Para os cálculos bidimensionais, o crescimento dendrítico é simulado para ligas ternárias de Fe-C-P. As alterações na concentração de fósforo afetam a velocidade da interface sólido/líquido. Essas alterações na velocidade da interface sólido/líquido se devem à baixa difusividade do fósforo durante o processo de solidificação.

Phase-field simulations of velocity selection in rapidly solidified binary alloys

Time-dependent simulations of two-dimensional isothermal Ni-Cu dendrites are simulated using a phase-field model solved with a finite-difference adaptive mesh refinement technique. Dendrite tip velocity selection is examined and found to exhibit a transition between two markedly different regimes as undercooling is increased. At low undercooling, the dendrite tip growth rate is consistent with the kinetics of the classical Stefan problem, where the interface is assume to be in local equilibrium. At high undercooling, the growth velocity selected approaches a linear dependence on melt undercooling, consistent with the continuous growth kinetics of Aziz and with a one-dimensional steady-state phase-field asymptotic analysis of Ahmad [Phys. Rev. E 58, 3436 (1998)]. Our simulations are also consistent with other previously observed behaviors of dendritic growth as undercooling is increased. These include the transition of dendritic morphology to absolute stability and nonequilibrium solute partitioning. Our results show that phase-field models of solidification, which inherently contain a nonzero interface width, can be used to study the dynamics of complex solidification phenomena involving both equilibrium and nonequilibrium interface growth kinetics.

Phase field modeling of solidification under stress

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.

Breakup of spiral and concentric ringed spherulites in polymer crystallization

Recently, the breakup of spiral and concentric ringed spherulites has been observed experimentally in isothermal crystallization of poly(vinylidene fluoride) in its blends with ethylene vinyl acetate copolymer. To elucidate the phenomenon of spiral breakup in polymer spherulites, a theoretical simulation has been undertaken in the framework of the phase field model of solidification based on time-dependent Ginzburg-Landau equations (TDGL model C) pertaining to the conserved concentration and nonconserved phase field order parameters. The concentration order parameter was described in terms of the Landau-type potential, while the crystal order parameter was expressed in terms of an asymmetric double-well potential to account for the nonequilibrium nature (i.e., metastability) of polymer crystallization. After inserting these potentials into the TDGL model C, the resulting coupled time-evolution equations show the emergence of the target and spiral patterns that undergo fragmentation at a very shallow supercooling. It is concluded that the depth of supercooling is the major contributor to the formation of the spirals and subsequent breakup of the spiral arms.

Crystal-liquid crystal binary phase diagrams

We propose a new theoretical scheme for the binary phase diagrams of crystal-liquid crystal mixtures by a combination of a phase field model of solidification, the Flory-Huggins theory for liquid-liquid mixing and Maier-Saupe-McMillan (FH-MSM) model for nematic and smectic liquid crystal orderings. The phase field theory describes the crystal phase transition of anisotropic organic crystal and/or side chain liquid crystalline polymer crystals while the FH-MSM model explains isotropic, nematic and smectic-A phase transitions. Self-consistent calculations reveal several possible phase diagram topologies of the binary crystal-liquid crystal mixtures. The calculated phase diagrams were found to accord well to the reported experimental results.

The effect of the ratio of solid to liquid conductivity on the side-branching characteristics of dendrites within a phase-field model of solidification

We use a phase-field model of dendritic growth in a pure undercooled melt to examine the effect of the ratio of the thermal conductivities in the solid and liquid states ( μ = κ s/ κ l) on the side-branching characteristics of the dendrite. We find that high conductivity in the solid favours extensive side-branching while low conductivity in the solid appears to strongly suppress side branching. Over the range 0.5 ⩽ μ ⩽ 2.0, which is typical of most (metallic) systems, which display dendritic growth the RMS distance at which the mean amplitude of the side-branches becomes equal to the tip radius varies from as little as 10 tip radii to in excess of 45 tip radii. This implies that there may be significant morphological difference between dendrites grown in different materials. The variation does not appear to follow exactly the analytical relationship predicted by solvability theory.

Sharp-interface projection of a fluctuating phase-field model

We present a derivation of the sharp-interface limit of a generic fluctuating phase-field model for solidification. As a main result, we obtain a sharp-interface projection which presents noise terms in both the diffusion equation and in the moving boundary conditions. The presented procedure does not rely on the fluctuation-dissipation theorem, and can therefore be applied to account for both internal and external fluctuations in either variational or nonvariational phase-field formulations. In particular, it can be used to introduce thermodynamical fluctuations in nonvariational formulations of the phase-field model, which permit to reach better computational efficiency and provide more flexibility for describing some features of specific physical situations. This opens the possibility of performing quantitative phase-field simulations in crystal growth while accounting for the proper fluctuations of the system.

Modeling the formation and dynamics of polycrystals in 3D

Phase field models of solidification have been extended to include grain boundaries, using a variety of techniques. A model developed by Kobayashi et al. [Physica D 140 (2000) 141] has been used to model a host of physical phenomena, but has so far been confined to two dimensions. In this letter we describe how to extend this model of polycrystalline solidification to three dimensions (3D).

Suppression of Anomalous Interface Effects by Localization of Solute Redistribution in Thin Interface Phase-Field Modeling of Solidification

A phase field model for alloy solidification was developed to suppress the anomalous interface effects such as enhanced surface diffusion, chemical potential jump and surface stretching by localizing the solute redistribution into a narrow region within the phase-field interface. Application of this model to a free dendritic growth in an undercooled liquid yields quantitatively the same results as previously reported anti-trapping model. By localization of the solute redistribution into a region of single grid spacing the anomalous interfacial effects can be effectively suppressed. This model can be used for quantitative phase field calculation with an enhanced computational efficiency.

Phase field modeling of dendritic growth with high anisotropy

A phase field model of solidification with a reduced interface diffuseness was extended to crystal growth with sharp corners by using a regularization technique of gradient energy coefficient. The model was applied to simulate the highly anisotropic dendritic solidification in an undercooled pure material (symmetric model) and in a supersaturated alloy (one-sided model). Simulation was performed under a wide range of anisotropy covering the critical anisotropy, where corners appear in the equilibrium crystal shape. The growth rate of dendrite increased monotonically with increase in the anisotropy when the anisotropy strength was lower than the critical value. As the anisotropy crossed the critical value, however, the growth rate dropped down by about 10%, and then increased again with higher anisotropy. The growth velocity reached a maximum value, followed by a slight decrease with further increase in the anisotropy. Overall behavior of the dendritic growth with corners seemed to be understood by considering the decreasing range of the allowed interface orientation with increase in the anisotropy beyond the critical value.