An introduction to phase-field modeling of microstructure evolution
Introduction
Most materials are heterogeneous on the mesoscale. Their microstructure consists of grains or domains, which differ in structure, orientation and chemical composition. The physical and mechanical properties on the macroscopic scale highly depend on the shape, size and mutual distribution of the grains or domains. It is, therefore, extremely important to gain insight in the mechanisms of microstructure formation and evolution. However extensive theoretical and experimental research are hereto required, as microstructure evolution involves a large diversity of often complicated processes. Moreover, a microstructure is inherently a thermodynamic unstable structure that evolves in time. Within this domain, the phase-field method has become a powerful tool for simulating the microstructural evolution in a wide variety of material processes, such as solidification, solid-state phase transformations, precipitate growth and coarsening, martensitic transformations and grain growth.
The microstructures considered in phase-field simulations typically consist of a number of grains. The shape and mutual distribution of the grains is represented by functions that are continuous in space and time, the phase-field variables. Within the grains, the phase-field variables have nearly constant values, which are related to the structure, orientation and composition of the grains. The interface between two grains is defined as a narrow region where the phase-field variables gradually vary between their values in the neighboring grains. This modeling approach is called a diffuse-interface description. The evolution of the shape of the grains, or in other words the position of the interfaces, as a function of time, is implicitly given by the evolution of the phase-field variables. An important advantage of the phase-field method is that, thanks to the diffuse-interface description, there is no need to track the interfaces (to follow explicitly the position of the interfaces by means of mathematical equations) during microstructural evolution. Therefore, the evolution of complex grain morphologies, typically observed in technical alloys, can be predicted without making any a priori assumption on the shape of the grains. The temporal evolution of the phase-field variables is described by a set of partial differential equations, which are solved numerically. Different driving forces for microstructural evolution, such as a reduction in bulk energy, interfacial energy and elastic energy, can be considered. The phase-field method has a phenomenological character: the equations for the evolution of the phase-field variables are derived based on general thermodynamic and kinetic principles; however, they do not explicitly deal with the behavior of the individual atoms. As a consequence, material specific properties must be introduced into the model through phenomenological parameters that are determined based on experimental and theoretical information.
Nowadays, the phase-field technique is very popular for simulating processes at the mesoscale level. The range of applicability is growing quickly, amongst other reasons because of increasing computer power. Besides solidification [1] and solid-state phase transformations [2], phase-field models are applied for simulating grain growth [3], dislocation dynamics [4], [5], [6], crack propagation [7], [8], electromigration [9], solid-state sintering [10], [11], [12] and vesicle membranes in biological applications [13], [14]. In current research, much attention is also given to the quantitative aspects of the simulations, such as parameter assessment and computational techniques.
The aim of the paper is to give a comprehensive introduction to phase-field modeling. The basic concepts are explained and illustrated with examples from the literature to show the possibilities of the technique. Numerous references for further reading are indicated.
Section snippets
Historical evolution of diffuse-interface models
More than a century ago, van der Waals [15] already modeled a liquid–gas system by means of a density function that varies continuously at the liquid–gas interface. Approximately 50 years ago, Ginzburg and Landau [16] formulated a model for superconductivity using a complex valued order parameter and its gradients, and Cahn and Hilliard [17] proposed a thermodynamic formulation that accounts for the gradients in thermodynamic properties in heterogeneous systems with diffuse interfaces. The
Sharp-interface versus diffuse-interface models
There is a wide variety of phase-field models, but common to all is that they are based on a diffuse-interface description. The interfaces between domains are identified by a continuous variation of the properties within a narrow region (Fig. 1a), which is different from the more conventional approaches for microstructure modeling as for example used in DICTRA.1
In conventional
Phase-field variables
In the phase-field method, the microstructural evolution is analyzed by means of a set of phase-field variables that are continuous functions of time and spatial coordinates. A distinction is made between variables related to a conserved quantity and those related to a non-conserved quantity. Conserved variables are typically related to the local composition. Non-conserved variables usually contain information on the local (crystal) structure and orientation. The set of phase-field variables
Thermodynamic energy functional
The driving force for microstructural evolution is the possibility to reduce the free energy of the system. The free energy may consist of bulk free energy , interfacial energy , elastic strain energy and energy terms due to magnetic or electrostatic interactions The bulk free energy determines the compositions and volume fractions of the equilibrium phases. The interfacial energy and strain energy affect the equilibrium compositions and volume
Phase-field equations
In the phase-field method, the temporal evolution of the phase-field variables is given by a set of coupled partial differential equations, one equation for each variable. Except for a few solidification models that are only concerned with reproducing the traditional sharp-interface models, the equations are derived according to the principles of non-equilibrium thermodynamics [145]. They are chosen so that the free energy decreases monotonically and mass is conserved for all components.
Quantitative phase-field simulations for alloy development
The early phase-field simulations showed that the phase-field technique is a general and powerful technique for simulating the evolution of complex morphologies. Simulations could give important insights into the role of specific material or process parameters on the pattern formation in solidification and the shape and spatial distribution of precipitates or different orientation domains in solid-state phase transformations. However, the results were rather qualitative and there are two major
Summary
This paper gives an introduction to the phase-field method and an overview of its possibilities. The phase-field method is a versatile and powerful technique for simulating microstructural evolution, which is currently very popular. Amongst others, it has been applied to solidification, precipitate growth and coarsening, martensitic transformations and grain growth and, more recently, also to other solid-state phase transformations like the austenite to ferrite transformation in steels,
Acknowledgements
This text is based on parts of the doctoral thesis of Nele Moelans (K.U. Leuven, May 2006, promotors Bart Blanpain and Patrick Wollants). The doctoral research was granted by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT — Vlaanderen). From September 2006, Nele Moelans has been a Postdoctoral Fellow of the Research Foundation — Flanders (FWO — Vlaanderen). We thank both institutions for financial support. We also gratefully thank the other members
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