Elsevier

Calphad

Volume 32, Issue 2, June 2008, Pages 268-294
Calphad

An introduction to phase-field modeling of microstructure evolution

https://doi.org/10.1016/j.calphad.2007.11.003Get rights and content

Abstract

The phase-field method has become an important and extremely versatile technique for simulating microstructure evolution at the mesoscale. Thanks to the diffuse-interface approach, it allows us to study the evolution of arbitrary complex grain morphologies without any presumption on their shape or mutual distribution. It is also straightforward to account for different thermodynamic driving forces for microstructure evolution, such as bulk and interfacial energy, elastic energy and electric or magnetic energy, and the effect of different transport processes, such as mass diffusion, heat conduction and convection. The purpose of the paper is to give an introduction to the phase-field modeling technique. The concept of diffuse interfaces, the phase-field variables, the thermodynamic driving force for microstructure evolution and the kinetic phase-field equations are introduced. Furthermore, common techniques for parameter determination and numerical solution of the equations are discussed. To show the variety in phase-field models, different model formulations are exploited, depending on which is most common or most illustrative.

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 F may consist of bulk free energy Fbulk, interfacial energy Fint, elastic strain energy Fel and energy terms due to magnetic or electrostatic interactions FfysF=Fbulk+Fint+Fel+Ffys. 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

References (203)

  • A. Artemev et al.

    Three-dimensional phase field model of proper martensitic transformation

    Acta Mater.

    (2001)
  • S. Choudhury et al.

    Phase-field simulation of polarization switching and domain evolution in ferroelectric polycrystals

    Acta Mater.

    (2005)
  • J. Zhang et al.

    Phase-field microelasticity theory and micromagnetic simulations of domain structures in giant magnetostrictive materials

    Acta Mater.

    (2005)
  • Y. Le Bouar et al.

    Origin of chessboard-like structures in decomposing alloys, theoretical model and computer simulation

    Acta Mater.

    (1998)
  • O. Penrose et al.

    Thermodynamically consistent models of phase-field type for the kinetics of phase transitions

    Physica D

    (1990)
  • S.-L. Wang et al.

    Thermodynamically-consistent phase-field models for solidification

    Physica D

    (1993)
  • R. Kobayashi

    Modeling and numerical simulations of dendritic crystal growth

    Physica D

    (1993)
  • I. Loginova et al.

    The phase-field approach and solute drag modeling of the transition to massive γα transformation in binary Fe–C alloys

    Acta Mater.

    (2003)
  • R. Tonhardt et al.

    Phase-field simulation of dendritic growth in a shear flow

    J. Cryst. Growth

    (1998)
  • I. Steinbach et al.

    A phase field concept for multiphase systems

    Physica D

    (1996)
  • J. Tiaden et al.

    The multiphase-field model with an integrated concept for modelling solute diffusion

    Physica D

    (1998)
  • R. Kobayashi et al.

    Vector-valued phase field model for crystallization and grain boundary formation

    Physica D

    (1998)
  • H. Kobayashi et al.

    Phase-field model for solidification of ternary alloys coupled with thermodynamic database

    Scr. Mater.

    (2003)
  • S. Hu et al.

    A phase-field model for evolving microstructures with strong elastic inhomogeneity

    Acta Mater.

    (2001)
  • K. Wu et al.

    A phase field study of microstructural changes due to the Kirkendall effect in two-phase diffusion couples

    Acta Mater.

    (2001)
  • W. Dreyer et al.

    A study of the coarsening in tin/lead solders

    Int. J. Solids Struct.

    (2000)
  • W. Dreyer et al.

    Modeling diffusional coarsening in eutectic tin/lead solders: A quantitative approach

    Int. J. Solids Struct.

    (2001)
  • S.M. Allen et al.

    A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening

    Acta Metall.

    (1979)
  • Y. Wang et al.

    Effect of antiphase domains on shape and spatial arrangement of coherent ordered intermetallics

    Scr. Metall. Mater.

    (1994)
  • V. Vaithyanathan et al.

    Coarsening of ordered intermetallic precipitates with coherency stress

    Acta Mater.

    (2002)
  • D. Li et al.

    Shape of a rhombohedral coherent Ti11Ni14 precipitate in a cubic matrix and its growth and dissolution during constrained aging

    Acta Mater.

    (1997)
  • Y. Wen et al.

    Effect of elastic interaction on the formation of a complex multi-domain microstructural pattern during a coherent hexagonal to orthorhombic transformation

    Acta Mater.

    (1999)
  • Y. Wen et al.

    Coarsening dynamics of self-accommodating coherent patterns

    Acta Mater.

    (2002)
  • L.-Q. Chen

    A novel computer simulation technique for modeling grain growth

    Scr. Metall. Mater.

    (1995)
  • D. Fan et al.

    Computer simulation of grain growth using a continuum field model

    Acta Mater.

    (1997)
  • D. Fan et al.

    Computer simulation of topological evolution in 2-D grain growth using a continuum diffuse-interface field model

    Acta Mater.

    (1997)
  • A. Kazaryan et al.

    Grain growth in anisotropic systems: Comparison of effects of energy and mobility

    Acta Mater.

    (2002)
  • N. Ma et al.

    Computer simulation of texture evolution during grain growth: Effect of boundary properties and initial microstructure

    Acta Mater.

    (2004)
  • N. Moelans et al.

    Pinning effect of second-phase particles on grain growth in polycrystalline films studied by 3-d phase field simulations

    Acta Mater.

    (2007)
  • D. Fan et al.

    Diffusion-controlled grain growth in two-phase solids

    Acta Mater.

    (1997)
  • C. Lan et al.

    Adaptive phase field simulation of non-isothermal free dendritic growth of a binary alloy

    Acta Mater.

    (2003)
  • J. Ramirez et al.

    Examination of binary alloy free dendritic growth theories with a phase-field model

    Acta Mater.

    (2005)
  • B. Nestler et al.

    Crystal growth of pure substances: Phase-field simulations in comparison with analytical and experimental results

    J. Comput. Phys.

    (2005)
  • C. Lan et al.

    Efficient adaptive phase field simulation of directional solidification of a binary alloy

    J. Cryst. Growth

    (2003)
  • J. Warren et al.

    Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method

    Acta Metall. Mater.

    (1995)
  • I. Loginova et al.

    On the formation of Widmanstätten ferrite in binary Fe–C — phase-field approach

    Acta Mater.

    (2004)
  • M. Mecozzi et al.

    Phase field modelling of the interfacial condition at the moving interphase during the γα transformation in C–Mn steels

    Comput. Mater. Sci.

    (2005)
  • B. Nestler et al.

    Phase-field modeling of multi-phase solidification

    Comput. Phys. Commun.

    (2002)
  • W.J. Boettinger et al.

    Phase-field simulation of solidification

    Ann. Rev. Mater. Res.

    (2002)
  • L.-Q. Chen

    Phase-field models for microstructure evolution

    Ann. Rev. Mater. Res.

    (2002)
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