Phase-field Variable Research Articles

Bounding box algorithm for three-dimensional phase-field simulations of microstructural evolution in polycrystalline materials

Phase-field modeling has proven to be a versatile tool for simulating microstructural evolution phenomena, such as grain growth in polycrystalline materials. However, the computing time and computing memory requirements of a phase-field model pose severe limitations on the number of phase-field variables that can be taken into account in a practical implementation. In this paper, a sparse bounding box algorithm is proposed that allows the use of a large number of phase-field variables without excessive memory usage or computational requirements. The algorithm is applied to a three-dimensional model for grain growth in the presence of second-phase particles.

Phase Field Modeling of Hysteresis in Sessile Drops

We propose a novel approach to describe wetting of plane solid surfaces by liquid drops. A two-dimensional nonconserved phase field variable is employed to distinguish between wetted and nonwetted regions on the surface. The imbalance in the Young's force provides for the exchange of relative stability of the two phases. The three-phase contact line tension arises from the gradient energy and contact angle hysteresis from the kinetic coefficient. Using this theory, we discuss contact angle hysteresis on chemically heterogeneous surfaces. We show significant departure from the classical Cassie theory, which is attributed to defect pinning of the continuous triple line.

Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing

This paper describes a novel numerical algorithm for simulating interfacial dynamics of non-Newtonian fluids. The interface between two immiscible fluids is treated as a thin mixing layer across which physical properties vary steeply but continuously. The property and evolution of the interfacial layer is governed by a phase-field variable ϕ that obeys a Cahn–Hilliard type of convection-diffusion equation. This circumvents the task of directly tracking the interface, and produces the correct interfacial tension from the free energy stored in the mixing layer. Viscoelasticity and other types of constitutive equations can be incorporated easily into the variational phase-field framework. The greatest challenge of this approach is in resolving the sharp gradients at the interface. This is achieved by using an efficient adaptive meshing scheme governed by the phase-field variable. The finite-element scheme easily accommodates complex flow geometry and the adaptive meshing makes it possible to simulate large-scale two-phase systems of complex fluids. In two-dimensional and axisymmetric three-dimensional implementations, the numerical toolkit is applied here to drop deformation in shear and elongational flows, rise of drops and retraction of drops and torii. Some of these solutions serve as validation of the method and illustrate its key features, while others explore novel physics of viscoelasticity in the deformation and evolution of interfaces.

Efficient numerical algorithm for multiphase field simulations

Phase-field models have emerged as a successful class of models in a wide variety of applications in computational materials science. Multiphase field theories, as a subclass of phase-field theories, have been especially useful for studying nucleation and growth in polycrystalline materials. In theory, an infinite number of phase-field variables are required to represent grain orientations in a rotationally invariant free energy. However, limitations on available computational time and memory have restricted the number of phase-field variables used in the simulations. We present an approach by which the time and memory requirements are drastically reduced relative to standard algorithms. The proposed algorithm allows us the use of an unlimited number of phase-field variables to perform simulations without the associated burden on computational time or memory. We present the algorithm in the context of coalescence free grain growth.

PDE's on surfaces---a diffuse interface approach

We introduce a new approach to deal with the numerical solution of partial differential equations on surfaces. Thereby we reformulate the problem on a larger domain in one higher dimension and introduce a diffuse interface region of a phase-field variable, which is defined in the whole domain. The surface of interest is now only implicitly given by the $1=2$-level set of this phase-field variable. Formal matched asymptotics show the convergence of the reformulated problem to the original PDE on the surface, as the diffuse interface width shrinks to zero. The main advantage of the approach is the possibility to formulate the problem on a Cartesian grid. With adaptive grid refinement the additional computational cost resulting from the higher dimension can be significantly reduced. Examples on linear diffusion and nonlinear phase separation demonstrate the wide applicability of the method.

A phase field model for the simulation of grain growth in materials containing finely dispersed incoherent second-phase particles

A phase field model is presented for simulating grain growth in materials containing small incoherent second-phase particles that are constant in time. The model of [L.-Q. Chen, W. Yang, Phys. Rev. B 50 (1994) 15752] for grain growth in single phase materials, that represents a polycrystalline microstructure with a set of phase field variables η i ( r, t), has been modified to incorporate the presence of second-phase particles by adding an extra term ϵ Φ ∑ i η i 2 to the free energy expression, with Φ a spatially dependent parameter that equals one inside the particles, and zero elsewhere and ϵ a positive constant. The interaction between a particle and a diffuse interphase is analyzed from an energetic point of view and the effect of pinning particles on the shape and the evolution of an initially spherical grain is studied for two-dimensional and three-dimensional systems.

Phase-field simulations of solidification in binary and ternary systems using a finite element method

We present adaptive finite element simulations of dendritic and eutectic solidification in binary and ternary alloys. The computations are based on a recently formulated phase-field model that is especially appropriate for modelling non-isothermal solidification in multicomponent multiphase systems. In this approach, a set of governing equations for the phase-field variables, for the concentrations of the alloy components and for the temperature has to be solved numerically, ensuring local entropy production and the conservation of mass and inner energy. To efficiently perform numerical simulations, we developed a numerical scheme to solve the governing equations using a finite element method on an adaptive non-uniform mesh with highest resolution in the regions of the phase boundaries. Simulation results of the solidification in ternary Ni 60 Cu 40 - x Cr x alloys are presented investigating the influence of the alloy composition on the growth morphology and on the growth velocity. A morphology diagram is obtained that shows a transition from a dendritic to a globular structure with increasing Cr concentrations. Furthermore, we comment on 2D and 3D simulations of binary eutectic phase transformations. Regular oscillatory growth structures are observed combined with a topological change of the matrix phase in 3D. An outlook for the application of our methods to describe AlCu eutectics is given.

Phase-field model for island dynamics in epitaxial growth

A phase-field model is proposed to describe epitaxial growth. In this model the motion of island boundaries of discrete atomic layers is determined by the time evolution of an introduced phase-field variable. We use formally matched asymptotic expansion to determine the asymptotic limit of vanishing interfacial thickness and show the reduction to classical sharp interface models of Burton–Cabrera–Frank type.

Various phase-field approximations for Epitaxial Growth

We present diffuse interface approximations for a step flow model in epitaxial growth. In this model, the motion of step edges of discrete atomic layers is determined by the time evolution of an introduced phase-field variable. In order to incorporate the attachment–detachment kinetics at step edges into the phase-field model a degenerate mobility-function is established. The model is used to simulate the evolution of a step train.

Phase-Field Simulation of Solidification

▪ Abstract An overview of the phase-field method for modeling solidification is presented, together with several example results. Using a phase-field variable and a corresponding governing equation to describe the state (solid or liquid) in a material as a function of position and time, the diffusion equations for heat and solute can be solved without tracking the liquid-solid interface. The interfacial regions between liquid and solid involve smooth but highly localized variations of the phase-field variable. The method has been applied to a wide variety of problems including dendritic growth in pure materials; dendritic, eutectic, and peritectic growth in alloys; and solute trapping during rapid solidification.

An Adaptive Finite Volume Method for Incompressible Heat Flow Problems in Solidification

An adaptive finite volume method is presented for solving incompressible heat flow problems with an unknown melt/solid interface, mainly in solidification applications, using primitive variables on a fixed collocated grid. A phase-field variable is introduced to treat the melt/solid interface, which is assumed to be diffusive, so that the complicated interfaces and phase change (using the enthalpy model) can be treated easily. The method is implemented through an object-oriented way based on adaptive mesh refinement and coarsening using dynamic data structures and derived data types of FORTRAN90. In addition to the refinement on the interfaces or boundaries, the mesh can be adapted to a solution based on numerical errors or gradients. Extensive tests are performed for cases with a fixed or free interface, and excellent agreement with the body-fitted or front tracking schemes is obtained. Furthermore, by gradual reduction of the interface thickness, the sharp-interface limit can be reached, which ensures the correctness of using a diffusive interface. The present approach is particularly suitable for problems having a complicated interface morphology as well as phase evolution, such as the phase-field simulation of dendritic growth. Two examples, without and with convection, are further given and good agreement with previous results are found.

Computer Modeling of the Evolution of Dendrite Microstructure in Binary Alloys During Non-isothermal Solidification

Two-dimensional simulations of the evolution of dendrite microstructure during isothermal and non-isothermal solidifications of a Ni-0.41Cu binary alloy are carried out using the phase-field method. The governing evolution equation for the phase field variable, the solute mole fraction and the temperature are formulated and numerically solved using an explicit finite difference scheme. To make the computations tractable, parallel computing is employed. The results obtained show that under lower cooling rates, the solidification process is controlled by partitioning of the solute between the solid and the liquid at the solid/liquid interface. At high cooling rates, on the other hand, solute trapping takes place and solidification is controlled by the heat extraction rate. An increase in the cooling rate is also found to have a pronounced effect on the dendrite microstructure causing it to change from poorly developed dendrites consisting of only primary stalks, via fully developed dendrites containing secondary and tertiary arms to the diamond-shaped grains with cellular surfaces. These findings are in excellent agreement with experimental observations.

Phase-field model of solidification of a binary alloy

We develop a rather general thermodynamically consistent phase-field model for solidification of a binary alloy, based on an entropy functional that contains squared gradient terms in the energy density, the composition and the phase-field variable. By assuming positive local entropy production, we derive generalized phase-field equations for an alloy, including cross terms that connect thermal and compositional driving forces to energy and solute fluxes. We explore this model in detail for a regular solution and show that four existing models can be recovered as special cases. We also use it to develop a new phase-field model for an alloy in which an explicit phase-field variable is absent.