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3、eviers archiving and manuscript policies areencouraged to visit:http:/ personal copyComputer Coupling of Phase Diagrams and Thermochemistry 32 (2008) introduction to phase-field modeling of microstructure evolutionNele Moelans, Bart Blanpain, Patrick WollantsDepartment of Metallurgy and Materials E
4、ngineering, Faculty of Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 44, B-3001 Leuven, BelgiumReceived 1 June 2007; received in revised form 10 October 2007; accepted 1 November 2007Available online 21 December 2007AbstractThe phase-field method has become an important and extre
5、mely 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 ontheir shape or mutual distribution. It is also straightforward to account
6、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 massdiffusion, heat conduction and convection. The purpose of the paper is to give an
7、introduction to the phase-field modeling technique. The conceptof diffuse interfaces, the phase-field variables, the thermodynamic driving force for microstructure evolution and the kinetic phase-field equationsare introduced. Furthermore, common techniques for parameter determination and numerical
8、solution of the equations are discussed. To show thevariety in phase-field models, different model formulations are exploited, depending on which is most common or most illustrative.c 2007 Elsevier Ltd. All rights reserved.Keywords: Phase-field modeling; Microstructure; Nonequilibrium thermodynamics
9、; Kinetics; Simulation1. IntroductionMost materials are heterogeneous on the mesoscale. Theirmicrostructure consists of grains or domains, which differ instructure, orientation and chemical composition. The physicaland mechanical properties on the macroscopic scale highlydepend on the shape, size an
10、d mutual distribution of thegrains or domains. It is, therefore, extremely important togain insight in the mechanisms of microstructure formationand evolution. However extensive theoretical and experimentalresearch are hereto required, as microstructure evolutioninvolves a large diversity of often c
11、omplicated processes.Moreover, a microstructure is inherently a thermodynamicunstable structure that evolves in time. Within this domain,the phase-field method has become a powerful tool forsimulating the microstructural evolution in a wide variety ofmaterial processes, such as solidification, solid
12、-state phasetransformations, precipitate growth and coarsening, martensitictransformations and grain growth.The microstructures considered in phase-field simulationstypically consist of a number of grains. The shape and mutual Corresponding author. Tel.: +32 16 321278; fax: +32 16 321991.E-mail addr
13、ess: nele.moelansmtm.kuleuven.be (N. Moelans).URL: http:/nele.studentenweb.org (N. Moelans).distribution of the grains is represented by functions that arecontinuous in space and time, the phase-field variables. Withinthegrains,thephase-fieldvariableshavenearlyconstantvalues,which are related to the
14、 structure, orientation and compositionof the grains. The interface between two grains is definedas a narrow region where the phase-field variables graduallyvary between their values in the neighboring grains. Thismodeling approach is called a diffuse-interface description.The evolution of the shape
15、 of the grains, or in other wordsthe position of the interfaces, as a function of time, isimplicitly 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 needto track the interfaces (to f
16、ollow explicitly the position ofthe interfaces by means of mathematical equations) duringmicrostructural evolution. Therefore, the evolution of complexgrain morphologies, typically observed in technical alloys, canbe predicted without making any a priori assumption on theshape of the grains. The tem
17、poral evolution of the phase-field variables is described by a set of partial differentialequations, which are solved numerically. Different drivingforces for microstructural evolution, such as a reduction inbulk energy, interfacial energy and elastic energy, can beconsidered. The phase-field method
18、 has a phenomenological0364-5916/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.calphad.2007.11.003Authors personal copyN. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294 269character: the equations for the evolution of the phase
19、-fieldvariables are derived based on general thermodynamic andkinetic principles; however, they do not explicitly deal with thebehavior of the individual atoms. As a consequence, materialspecific properties must be introduced into the model throughphenomenological parameters that are determined base
20、d onexperimental and theoretical information.Nowadays, the phase-field technique is very popular forsimulating processes at the mesoscale level. The range ofapplicability isgrowingquickly,amongstotherreasonsbecauseof increasing computer power. Besides solidification 1 andsolid-state phase transforma
21、tions 2, phase-field models areapplied for simulating grain growth 3, dislocation dynamics46, crack propagation 7,8, electromigration 9, solid-state sintering 1012 and vesicle membranes in biologicalapplications 13,14. In current research, much attention is alsogiven to the quantitative aspects of t
22、he simulations, such asparameter assessment and computational techniques.The aim of the paper is to give a comprehensive introductionto phase-field modeling. The basic concepts are explainedand illustrated with examples from the literature to show thepossibilities of the technique. Numerous referenc
23、es for furtherreading are indicated.2. Historical evolution of diffuse-interface modelsMore than a century ago, van der Waals 15 alreadymodeled a liquidgas system by means of a densityfunction that varies continuously at the liquidgas interface.Approximately 50 years ago, Ginzburg and Landau 16formu
24、lated a model for superconductivity using a complexvalued order parameter and its gradients, and Cahn andHilliard 17 proposed a thermodynamic formulation thataccounts for the gradients in thermodynamic properties inheterogeneous systems with diffuse interfaces. The stochastictheory of critical dynam
25、ics of phase transformations fromHohenberg and Halperin 18 and Gunton et al. 19 also resultsin equations that are very similar to the current phase-fieldequations. Nevertheless, the concept of diffuse interfaces wasintroduced into microstructural modeling only 20 years ago.There are essentially two
26、types of phase-field model that havebeen developed independently by two communities.The first type of phase-field model was derived byChen 20 and Wang 21 from the microscopic theory ofKhachaturyan 22,23. The phase-field variables are relatedto microscopic parameters, such as the local composition an
27、dlong-range order parameter fields reflecting crystal symmetryrelations between coexisting phases. The model has beenapplied to a variety of solid-state phase transformations(see 2 for an overview) that involve a symmetry reduction,for example the precipitation of an ordered intermetallicphase from
28、a disordered matrix 2426 and martensitictransformations 27,28. It was also applied to ferroelectric29,30 and magnetic domain evolution 31 and can accountfor the influence of elastic strain energy on the evolution ofthe microstructure. The group of Chen at Pennsylvania StateUniversity (USA), the grou
29、p of Wang at Ohio State University(USA) and the group of Khachaturyan at Rutgers University(USA) are leading within this phase-field community. Similarphase-field models are used by Miyazaki 32,33 (NagoyaInstitute of Technology, Japan) and Onuki and Nishimori 34(Kyoto University, Japan) to describe
30、spinodal decompositionin materials with a composition-dependent molar volume andby Finel and Le Bouar (ONERA, France) for describing theinteraction of stress, strain and dislocations with precipitategrowth and structural phase transitions 5,35,36.Inthesecondtypeofphase-field model, aphenomenological
31、phase-field is used purely to avoid tracking the interface.The idea was introduced by Langer 37 based on oneof the stochastic models of Hohenberg and Halperin 18.The model is mainly applied to solidification, for exampleto study the growth of complex dendrite morphologies,the microsegregation of sol
32、ute elements and the coupledgrowth in eutectic solidification (see 1,38 for an overview).Important contributions were, amongst others, due to Caginalp(University of Pittsburgh, USA) 39,40, Penrose and Fife 41,Wang and Sekerka 42, Kobayashi 43, Wheeler, Boettingerand McFadden (NIST, USA and Universit
33、y of Southampton,UK) 4448, Kim and Kim (Kunsan National Universityand Chongju University, Korea) 49, Karma (NortheasternUniversity, USA) and Plapp (Ecole Polytechnique, France)5054andattheRoyalInstituteofTechnology(Sweden)5557. Multiphase-field models for systems with more than twocoexisting phases
34、were formulated by Steinbach et al. 58,59 (Access, Germany) and Nestler (Karlsruhe University ofApplied Sciences, Germany), Garcke and Stinner (Universityof Regensburg, Germany) 60,61. Vector-valued phase-fieldmodelsinwhichthephase-fieldiscombinedwithanorientationfield for representing different cry
35、stal or grain orientations havebeen developed by Kobayashi (Hokkaido University, Japan),Warren (NIST, USA) and Carter (MIT, USA) 62,63 and byGranasy et al. (Research institute for solid state physics andoptics, Hungary) 64.3. Sharp-interface versus diffuse-interface modelsThere is a wide variety of
36、phase-field models, but commonto all is that they are based on a diffuse-interface description.The interfaces between domains are identified by a continuousvariation of the properties within a narrow region (Fig. 1a),which is different from the more conventional approaches formicrostructure modeling
37、 as for example used in DICTRA.1In conventional modeling techniques for phase transforma-tions and microstructural evolution, the interfaces between dif-ferent domains are considered to be infinitely sharp (Fig. 1b),and a multi-domain structure is described by the position ofthe interfacial boundari
38、es. For each domain, a set of differentialequations is solved along with flux conditions and constitutivelaws at the interfaces. For example, in the case of diffusion con-trolledgrowthofonephaseattheexpense ofanother phase,1 Software for simulating DIffusion Controlled phase TRAnsformationsfrom Ther
39、mo-Calc Software (http:/www.thermocalc.se).Authors personal copy270 N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294Fig. 1. (a) Diffuse interface: properties evolve continuously between theirequilibrium values in the neighboring grains. (b) Sharp interface
40、: propertiesare discontinuous at the interface.a diffusion equation for the solute concentration c (mol/m3) issolved in each phase domainct = D2c, for phase (1)ct = D2c, for phase (2)subject to a flux balance that expresses solute conservation atthe interface(c,int c,int)= Dcr1 Dcr1 (3)and the const
41、itutional requirement that both phases are inthermodynamic equilibrium at the interface(c,int)=(c,int). (4)c and c are the molar concentrations of the soluteelement in respectively the -phase and the -phase, Dand D the diffusion coefficients and c,int and c,int themolar concentrations at the interfa
42、ce. and representthe chemical potentials of the solute for respectively the-phase and the -phase. r1 is the spatial coordinate alongthe axis perpendicular to the interface. This sharp-interfacemethodology requires explicit tracking of the moving interfacebetween the -phase and the -phase, which is e
43、xtremelydifficult from a mathematical point of view for the complexgrain morphologies in real alloys. Sharp-interface simulationsare mostly restricted to one-dimensional systems or simplifiedgrain morphologies that, for example, consist of sphericalgrains.In the diffuse-interface approach, the micro
44、structure isrepresented by means of a set of phase-field variables that arecontinuous functions of space and time. Within the domains,the phase-field variables have the same values as in the sharp-interface model (see Fig. 1a). However, the transition betweenthese values at interfaces is continuous.
45、 The position of theinterfaces is thus implicitly given by a contour of constantvalues of the phase-field variables and the kinetic equations formicrostructural evolution are defined over the whole system.Using a diffuse-interface description, it is possible to predictthe evolution of complex grain
46、morphologies as well as atransition in morphology, like the splitting or coalescence ofprecipitates and the transition from cellular to dendritic growthin solidification. Furthermore, no constitutional relations ofthe form (4) are imposed at the interfaces. Therefore, non-equilibrium effects at the
47、moving interfaces, like solute dragand solute trapping, can be studied as a function of the velocityof the interface. Flux conditions are implicitly considered in thekinetic equations.4. Phase-field variablesIn the phase-field method, the microstructural evolutionis analyzed by means of a set of pha
48、se-field variables thatare continuous functions of time and spatial coordinates.A distinction is made between variables related to aconserved quantity and those related to a non-conservedquantity. Conserved variables are typically related to thelocal composition. Non-conserved variables usually cont
49、aininformation on the local (crystal) structure and orientation. Theset of phase-field variables must capture the important physicsbehind the phase transformation or coarsening process. Sinceredundant variables increase the computational requirements,the number of variables is also best kept minimal.4.1. Composition variablesComposition variables like molar fractions or
