7 Phase-Field Modeling

~Brit ta Nestler

The following sections are devoted to introducing the phase-field modeling technique, numerical methods, and simulation applications to microstructure evolution and pattern formation in materials science. Model formulations and computations of pure substances and ofmulticomponent alloys are discussed. A thermodynamically consistent class of nonisothermal phase-field models for crystal growth and solidification in complex alloy systems is presented. Expressions for the different energy density contributions are proposed and explicit examples are given. Multicomponent diffusion in the bulk phases including interdiffusion coefficients as well as diffusion in the interfacial regions are formulated. Anisotropy of both, the surface energies and the kinetic coefficients, is incorporated in the model formulation. The relation of the diffuse interface models to classical sharp interface models by formally matched asymptotic expansions is summarized.

In Section 7.1, a motivation to develop phase-field models and a short historical background serve as an introduction to the topic, followed by a derivation of a first phase-field model for pure substances, that is, for solid-liquid phase systems in Section 7.2. On the basis of this model, we perform an extensive numerical case study to evaluate the individual terms in the phase-field equation in Section 7.3. The finite difference discretization methods, an implementation of the numerical algorithm, and an example of a concrete C++ program together with a visualiza- tion in MatLab is given. In Section 7.4, the extension of the fundamental phase-field model to describe phase transitions in multicomponent systems with multiple phases and grains is described. A 3D parallel simulator based on a finite difference discretization is introduced illus- trating the capability of the model to simultaneously describe the diffusion processes of multiple components, the phase transitions between multiple phases, and the development of the temper- ature field. The numerical solving method contains adaptive strategies and multigrid methods for optimization of memory usage and computing time. As an alternative numerical method, we also comment on an adaptive finite element solver for the set of evolution equations. Applying the computational methods, we exemplarily show various simulated microstructure formations in complex multicomponent alloy systems occurring on different time and length scales. In particular, we present 2D and 3D simulation results of dendritic, eutectic, and peritectic solidi- fication in binary and ternary alloys. Another field of application is the modeling of competing polycrystalline grain structure formation, grain growth, and coarsening.

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