Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294www.elsevier.com/locate/calphadAn introduction to phase-field modeling of microstructure evolution

Nele Moelans, Bart Blanpain, Patrick Wollants

Department of Metallurgy and Materials Engineering, Faculty of Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 44, B-3001 Leuven, Belgium

Received 1 June 2007; received in revised form 10 October 2007; accepted 1 November 2007Available online 21 December 2007

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 ontheir 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 massdiffusion, heat conduction and convection. The purpose of the paper is to give an 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 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; Kinetics; Simulation1. Introduction

Most 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 and 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 complicated 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-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 address: nele.moelans@mtm.kuleuven.be (N. Moelans).URL: http://nele.studentenweb.org (N. Moelans).

0364-5916/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.calphad.2007.11.003distribution of the grains is represented by functions that arecontinuous in space and time, the phase-field variables. Withinthe grains, the phase-field variables have nearly constant values,which are related to the 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 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 follow 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 temporal 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 has a phenomenological

http://www.elsevier.com/locate/calphadmailto:nele.moelans@mtm.kuleuven.behttp://nele.studentenweb.orghttp://nele.studentenweb.orghttp://nele.studentenweb.orghttp://nele.studentenweb.orghttp://dx.doi.org/10.1016/j.calphad.2007.11.003

N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294 269

1 Software for simulating DIffusion Controlled phase TRAnsformationsfrom Thermo-Calc Software (http://www.thermocalc.se).character: the equations for the evolution of the phase-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 based onexperimental and theoretical information.

Nowadays, the phase-field technique is very popular forsimulating processes at the mesoscale level. The range ofapplicability is growing quickly, amongst other reasons becauseof increasing computer power. Besides solidification [1] andsolid-state phase transformations [2], phase-field models areapplied for simulating grain growth [3], dislocation dynamics[46], 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 the 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 references for furtherreading are indicated.

2. Historical evolution of diffuse-interface models

More 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 [16]formulated 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 dynamics 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 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 andlong-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 a disordered matrix [2426] and martensitictransformations [27,28]. It was also applied to ferroelectric[29,30] 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 group 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 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].

In the second type of phase-field model, a phenomenologicalphase-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 solute 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 University of Southampton,UK) [4448], Kim and Kim (Kunsan National Universityand Chongju University, Korea) [49], Karma (NortheasternUniversity, USA) and Plapp (Ecole Polytechnique, France)[5054] and at the Royal Institute of Technology (Sweden) [5557]. Multiphase-field models for systems with more than twocoexisting phases 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-fieldmodels in which the phase-field is combined with an orientationfield for representing different crystal 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 models

There is a wide variety of 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 as for example used in DICTRA.1

In 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 boundaries. 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-trolled growth of one phase at the expense of another phase ,

http://www.thermocalc.se

270 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: propertiesare discontinuous at the interface.

a diffusion equation for the solute concentration c (mol/m3) issolved in each phase domain

c

t= D2c, for phase (1)

c

t= D2c , for phase (2)

subject to a flux balance that expresses solute conservation atthe interface

(c,int c,int) = Dc

r1 D

c

r1(3)

and the constitutional 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, D

and D the diffusion coefficients and c,int and c,int themolar concentrations at the interface. 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 extremelydifficult 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 microstructure 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. 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 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 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 variables

In the phase-field method, the microstructural evolutionis analyzed by means of a set of phase-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 containinformation 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 variables

Composition variables like molar fractions or concentrationsare typical examples of conserved properties, since the numberof moles of each component in the system is conserved. Assumea system with C components and ni , i = 1 . . .C , the numberof moles of each component i . Then the molar fraction xi andmolar concentration ci (mol/m3) of component i are defined as

xi =nintot

(5)

and

ci =niV=

xiVm, (6)

where ntot =

ni is the total number of moles in the system,Vm the molar volume and V the total volume of the system. Ateach position r in the system

Ci=1

xi = 1 enCi=1

ci =ntotV

=1Vm. (7)

As a consequence, the composition field is completely definedby C 1 molar fraction fields xi (r ), or C 1 molarconcentration fields ci (r ) in combination with the molarvolume Vm. Since in a closed system the total number of molesof each component is conserved, the temporal evolution ofeach molar fraction field xi (r ) or concentration field ci (r )is restricted by the integral relationVcidr =

1Vm

Vxidr = ni = ct. (8)

Changes in local composition can only occur by fluxes of atomsbetween neighboring volume elements. Other compositionvariables are mass fraction or mass per cent.

N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294 271Fig. 2. Representation of a microstructure consisting of two types of domainwith different composition, using a single molar fraction field xB (r ). Thefree energy density as a function of composition f0(xB ) has a concave part.Therefore, for a mean composition xB,0 in between x

1B and x

2B , the system

decomposes into two types of domain, one with composition x1B and the

other with composition x1B , in order to reduce its free energy content. The

compositions x1B and x2B are determined by the common tangent to the free

energy curve, and the free energy of the decomposed system corresponds to thepoints on the common tangent.

Fig. 2 shows the representation of a microstructure by asingle molar fraction field. The structure consists of two typesof domain with different compositions x1B and x

2B . At the

interface between two domains, xB(r ) varies smoothly butlocalized from x1B to x

2B . Such a representation was widely

used for spinodal decomposition and further coarsening ofthe decomposed structure [65,34,66,67,32,33] and precipitationand growth of precipitates [68,69] with the same structure as thematrix phase. A representation by means of composition fieldsonly is also applicable for systems containing domains with adifferent structure, if the microstructural evolution is diffusioncontrolled. This was for example the case in phase-fieldsimulations for ternary diffusion couples [70] and simulationsfor the coarsening of tin/lead solders [71,72].

4.2. Order parameters and phase-fields

Order parameters and phase-fields are both non-conservedvariables that are used to distinguish coexisting phaseswith a different structure. Order parameters refer to crystalsymmetry relations between coexisting phases. Phase-fields arephenomenological variables used to indicate which phase ispresent at a particular position in the system.

4.2.1. Order parametersThe concept of order parameters originates from micro-

scopic theories and was introduced in continuous theories byLandau for the description of phase transformations that involvea symmetry reduction [73]. Originally, the Landau theory wasdeveloped for the description of second-order phase transfor-mations at a critical temperature Tc.2 Each phase is represented2 Since, it has been proven that the Landau theory cannot describe the criticalphenomena occurring around Tc for a second-order phase transformation.Nevertheless, statistical theories and experimental facts have confirmed thatoutside the critical range, the Landau theory is applicable for first- and second-order phase transformations. Especially, the symmetry concepts introduced byLandau are useful.Fig. 3. Two-dimensional representation of an anti-phase structure with cubicsymmetry by means of a single order parameter field (r ). The orderedstructure consists of two sublattices occupied by a different type of atom,represented by black and white dots. The free energy density as a function of has a maximum at = 0, corresponding to the disordered structure whichis unstable, and two minima with equal depth at = 1, corresponding to thetwo equivalent variants of the ordered structure.

by a specific value of an order parameter (or combination of val-ues of a set of order parameters). Subsequently, a free energy or Landau polynomial is formulated as a function of the orderparameters and the temperature. For T > Tc the Landau poly-nomial has minima at the order parameter values correspond-ing to the high temperature phase and for T < Tc at the orderparameter values corresponding to the low temperature phase.The number of order parameters and the equilibrium values ofthe order parameters reflect the symmetry relations between thephases.

In the phase-field models of Chen and Wang and otherfollowers of Khachaturyans microscopic theory, order param-eter fields are mostly used for representing a microstructure. InFig. 3 is illustrated how a single order parameter field (r ) isused to describe the evolution of an ordered system with anti-phase boundaries in two dimensions [74]. The composition ofthe alloy is assumed to be constant. As shown in the figure, theordered structure consists of two sublattices, each occupied bya different type of atom. The atoms can occupy the sublatticesin two energetically equivalent ways. The two variants of theordered phase are distinguished by means of an order parameter. Within the ordered domains equals 1 or 1, depending onwhich sublattice is occupied by which type of atom. = 0 cor-responds to a disordered phase where the atoms are randomlydistributed over the sublattices.

Fig. 4 illustrates the use of order parameters for thedescription of a phase transformation that involves asymmetry reduction, in this case a cubic to tetragonal phasetransformation. The tetragonal phase has a lower symmetry

272 N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294Fig. 4. As a tetragonal structure has less symmetry than a cubic structure, it can be oriented in three equivalent ways with respect to the cubic structure. A set ofthree order parameter fields is used to distinguish the three tetragonal variants and the cubic phase. ac, at and ct are the crystal lattice parameters of the cubic andtetragonal phase.than the cubic phase. Therefore, it can have three orientationsthat are energetically equivalent (if only chemical energy isconsidered) and have the same probability to form from thecubic parent phase. Because of the transformation strainsassociated with a cubic to tetragonal transformation, domainswith a different variant of the tetragonal phase interact to reduceelastic strain energy. Therefore, it is relevant to use three orderparameters to distinguish the three variants. This representationwith three order parameter fields was for example used byWang [27] for modeling a cubic to tetragonal martensitictransformation.

To model the growth of ordered precipitates or precipitateswith a lower symmetry than the matrix phase, a set oforder parameter fields is combined with a molar fractionfield. The evolution of ordered precipitates with L12 structurein Ni-based superalloys (fcc structure) was, for example,frequently simulated using a phase-field model [75,24,25,76].In three dimensions, the ordered L12 structure consists of fourinterpenetrating simple cubic lattices which are all equivalentwith respect to the fcc disordered matrix phase. The fourvariants are distinguished by four combinations of three orderparameter fields, namely

(1, 1, 1)eq, (1,1, 1)eq, (1, 1,1)eq and

(1,1,1)eq.

Phase transformations with a more complicated symmetryreduction were also modeled, like the growth of rhombohedralTi11Ni14 precipitates in a cubic TiNi phase [77], the growth ofordered precipitates with tetragonal structure (L10 structure) ina cubic fcc matrix for CoPt and CuAuPt alloys [35], andstructural hexagonal to orthorhombic transformations [78,79].

The order parameter representation was generalized tomulti-domain structures by Chen [3,80]. The approach has beenused extensively for the study of grain growth [8187], wherethe different crystallographic orientations are represented witha large number of order parameter fields 1, 2, . . . , p. Withineach grain one of the field variables equals 1 or 1 and theothers equal 0. In this model the order parameter fields are nolonger related to crystallographic symmetry variants. They aremerely introduced to distinguish between different grains ordomains. The effect of solute drag on grain growth [88] andthe coarsening of two-phase structures [89,90] were consideredas well by adding an additional concentration field to the set oforder parameter fields.

There are no restrictions on the evolution of the orderparameter fields analogous with relations (7) and (8).

4.2.2. Phase-fieldsThe concept of a continuous non-conserved phase-field for

distinguishing two coexisting phases was introduced by Langer[37]. The phase-field equals 0 in one of the phases and 1 in theother. For example, in the case of solidification

= 0 in the liquid

= 1 in the solid, (9)

and varies continuously from 1 to 0 at the solidliquidinterface. Here, represents the local fraction of the solidphase. The single-phase-field representation in combinationwith a temperature and/or composition field was extensivelyapplied to study free dendritic growth [43,9193] in anundercooled melt, cellular pattern formation during directionalsolidification [53,94] and eutectic growth [47,95]. Non-equilibrium effects such as microsegregation [96] and solutetrapping [46,97] could be reproduced in the simulations.The concept of a phase-field was also applied to solid-state phase transformations, like the austenite to ferrite phasetransformation in steel (for example = 1 in the austenitephase and = 0 in the ferrite phase). The transitions betweendiffusion controlled transformation, Widmanstatten growth and

N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294 273massive transformation were investigated for an FeC alloy [56,98] and an FeCMn alloy [99].

The single-phase-field formalism was extended towardsmultiphase systems by Steinbach et al. [58,59]. In a multiphase-field model, a system with p coexisting phases is described bymeans of p phase-fields k , which represent the local fractionsof the different phases. Hence, the phase-fields have to sum upto one at every point r in the system

pk=1

k = 1 with k 0,k (10)

and only p 1 phase-fields are independent in a systemcontaining p coexisting phases. As phase-fields are non-conserved, there are no integral restrictions analogous withrelation (8) on the evolution of the phase-fields. The amountof a particular phase is in general not constant in time.

The multiphase-field models provide a general approachfor solidification reactions that involve multiple phases, suchas eutectic and peritectic solidification [100102,52]. Besidesthe general multiphase-field model, models with two [47] orthree [101] phase-fields were proposed specifically for eutecticreactions and a model with three phase-fields for coupled two-phase growth in general [54,103]. The multiphase-field modelwas also applied to grain growth in polycrystalline structures[104] and solid-state phase transformations such as pearliteformation in FeC [105,106].

5. Thermodynamic energy functional

The driving force for microstructural evolution is thepossibility to reduce the free energy of the system. The freeenergy F may consist of bulk free energy Fbulk, interfacialenergy Fint, elastic strain energy Fel and energy terms due tomagnetic or electrostatic interactions Ffys

F = Fbulk + Fint + Fel + Ffys. (11)

The bulk free energy determines the compositions and volumefractions of the equilibrium phases. The interfacial energy andstrain energy affect the equilibrium compositions and volumefractions of the coexisting phases, but also determine the shapeand mutual arrangement of the domains.

Different from classical thermodynamics where propertiesare assumed to be homogeneous throughout the system, the freeenergy F in the phase-field method is formulated as a functionalof the set of phase-field variables (which are functions of timeand spatial coordinates) and their gradients. When temperature,pressure and molar volume are constant and there are no elastic,magnetic or electric fields, the total free energy of a systemdefined by a concentration field xB and a set of order parameterfields k, k = 1 . . . p, is for example given by

F(xB, k) =V

f (xB, k, xB,

k)

=

V

[f0(xB, k)+

2( xB)

2+

k

k

2( k)

2

]dr .

(12)f0(xB, k) refers to a homogeneous system where all statevariables, in this case the phase-field variables, are constantthroughout the system. It is the classical free energyexpressed per unit of volume (J/m3) of a homogeneoussystem characterized by the local values of the phase-fieldvariables. f (xB, k,

xB,

k) is the local free energy

density of a heterogeneous systemwith diffuse interfaces. In thecase of classical Gibbsian thermodynamics for heterogeneous(i.e. multiphase) systems with sharp interfaces, the free energydensity would correspond to the common tangent to f0 for thesystem in Fig. 2 and to the minimum of f0 for the systemin Fig. 3. The gradient terms (/2)

xB and (k/2)

k are

responsible for the diffuse character of the interfaces. andk are called gradient energy coefficients. They are relatedto the interfacial energy and thickness. The gradient energycoefficients are always positive, so that gradients in the phase-field variables are energetically unfavorable and give rise tosurface tension. This free energy formalism for heterogeneoussystems with diffuse interfaces was first introduced by Cahnand Hilliard [17].

The homogeneous free energy density f0 is a function of thelocal values of the phase-field variables and reflects the equilib-rium bulk conditions of the coexisting domains. With respect tothe non-conserved phase-field variables, it has degenerate min-ima at the values the phase-fields or order parameters can havein the different domains (see for example the free energy func-tion in Fig. 3). For the conserved composition variables, thehomogeneous free energy has a common tangent at the equi-librium compositions of the coexisting phases (see for examplethe free energy function in Fig. 2).

According to Cahn and Hilliard [17], the interfacial energyFint of a system with diffuse interfaces and with a free energyfunctional of the form (12) is defined as

Fint(xB, k) =V

[1 f0(xB, k)+

2( xB)

2

+

k

k

2( k)

2

]dr , (13)

where 1 f0 is the difference between the bulk free energydensity for the actual local values of the phase-field variablesf0(xB, k) and the equilibrium free energy feq when interfacesare neglected (i.e. the equilibrium free energy accordingto classical Gibbsian thermodynamics for heterogeneoussystems). The meaning of 1 f0 is indicated in Figs. 2 and 3.Minimization of the integral (13) for a planar boundary givesthe equilibrium profile of the phase-field variables across aninterface and the specific interfacial energy.

Relation (13) shows that, in phase-field models, the inter-facial energy Fint consists of two contributions: one resultingfrom the fact that the phase-field variables differ from theirequilibrium values at the interfaces, and the other from the factthat interfaces are characterized by steep gradients in the phase-field variables. Within the domains 1 f0 and the gradient termsare equal to zero, since the phase-field variables are constantand have their equilibrium values. Therefore, bulk material does

274 N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294not contribute to the interfacial energy, as it is supposed to do.The width of the diffuse interfacial regions is the result of twoopposing effects. The wider, or the more diffuse, the interfa-cial region, the smaller the contribution of the gradient energyterms. However, for wider interfaces, there is more materialfor which the phase-field variables have non-equilibrium valuesand the integrated contribution from 1 f0 is accordingly larger.Therefore, larger values of the gradient energy coefficientsresult in more diffuse interfaces, whereas deeper energy wellsresult in sharper interfaces. For some phase-field formulations,there exist analytical relations between the gradient energy co-efficients and the interfacial energy and thickness. In general,however, interfacial energy must be evaluated numerically froman integral equation of the form 13.

Strictly speaking, Gibbs energy should be used in the caseof constant temperature and pressure and Helmholtz energy inthe case of constant temperature and volume. In many phase-field models, however, pressure and volume are assumed tobe constant. Under these restrictions, the Gibbs energy G andHelmholtz energy A only differ by a constant term, since G =A pV . Since

dG = dA d(pV ) = dA + 0, (14)

minimization of either of these functions leads to the sameconclusions. In general, the free energy F is used to describethe thermodynamic properties of the system in phase-fieldmodeling. It is not always stated clearly whether it refers toGibbs or Helmholtz energy.

Especially in the formulation of the free energy functional,there are many differences among the phase-field models. In thefollowing sections, a few examples of free energy functionalsare given. The models for solid-state phase transformationsinvolving coherency strains, the single-phase-field models andthe multiphase-field model are discussed.

5.1. Solid-state phase transformations with symmetry reduc-tion or ordering

The groups of Chen and Wang are specialized in phase-fieldsimulations for solid-state phase transformations involving asymmetry reduction or ordering reaction. It was illustrated inSection 4.2 that these phase transformations can be describedby physically well-defined order parameters that are related tothe crystal symmetry of the phases. The chemical contributionto the free energy of the system F is of the form (12), wherethe homogeneous free energy density f0 is expressed as aLandau polynomial of the order parameters and, if applicable,the composition. To incorporate the elastic energy producedby transformation strains, the standard phase-field model iscombined with the micro-elasticity theory of Khachaturyan,which allows us to express the elastic energy as a function ofcomposition and order parameters.

5.1.1. Landau expressions for the homogeneous free energydensity

In the following paragraphs, the Landau polynomial isformulated as a function of order parameters and compositionfor the structures introduced in Section 4.2.Anti-phase domain structure. By means of the anti-phasedomain structure depicted in Fig. 3, it is illustrated how theform of the Landau free energy function is determined. Usuallya fourth-order or sixth-order Landau polynomial is used. Themost general form of a polynomial of the fourth order for oneorder parameter is

f0() = fdis+ A + B2 + C3 + D4. (15)

Since both variants of the ordered structure are energeticallyequivalent, the free energy expression must be symmetricaround zero. Hence, all coefficients of uneven terms (A and C)are zero. If it is assumed that the free energy of the disorderedphase f dis is equal to zero and the minima are located at = 1and = 1, the following free energy expression is obtained:

f0() = 4(1 f0)max

(122 +

144), (16)

where (1 f0)max is the depth of the free energy. Since thefree energy has a maximum at = 0, the disordered phaseis unstable and will spontaneously evolve towards one of theordered variants.

Cubic to tetragonal transformation. In the case of the cubic totetragonal phase transformation in three dimensions depictedin Fig. 4, the Landau free energy as a function of three orderparameters must have six minima with equal depth at

(1, 2, 3) = (eq, 0, 0), (0,eq, 0), (0, 0,eq).

The value of eq depends on temperature and composition.Wang and Khachaturyan [27] proposed the following sixth-order expression for a cubic to tetragonal martensitictransformation:

f (1, 2, 3) =12A

3k=1

2i 14B

3k=1

4i

+16C

(3

k=1

2i

)3, (17)

with A, B and C positive constants that determine the value ofeq . Besides the degenerate minima corresponding with the sixsymmetry variants, it has a local minimum at (1, 2, 3) =(0, 0, 0) to express the first-order character of the martensitictransformation. Formation of a tetragonal domain within thecubic parent phase involves the formation of stable nuclei toovercome the energy barrier around (1, 2, 3) = (0, 0, 0).

Precipitation of an ordered intermetallic in a disordered matrix.The following Landau polynomial f0(xB, )

f0(xB, ) =12A(xB x

B)2+

12B(x B xB)

2

14C4 +

16D6 (18)

was proposed by Wang and Khachaturyan [75] for simulatingthe evolution of ordered precipitates in a disordered matrixfor a two-dimensional binary system. A, B,C, D are positive

N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294 275

3 The Landau expressions (16)(18) are, in fact, non-equilibrium freeenergies, since the free energy at thermodynamic equilibrium only depends onexternal state variables, such as the composition. The order parameters are so-called internal state variables, which describe internal degrees of freedom in asystem that is not in thermodynamic equilibrium [107]. The concept of internalstate variables is mainly applied for the description of phase transformationsand homogeneous reacting systems.Fig. 5. Landau free energy as a function of one composition variable xB andone order parameter as given by relation (18). The parameter values are takenfrom [75]. With respect to the order parameter, the Landau free energy hasa single minimum at = 0 for xB near x

disB and two degenerate minima

symmetrical around zero for xB near xorB .

phenomenological constants with a dimension of energy and x Band x B constants with a dimension of composition. In analogywith the antiphase domain structure in Fig. 3, it was assumedthat there are two equivalent variants of the ordered phase,represented by = eq and = eq. A three-dimensionalview of the free energy density as a function of the orderparameter and the molar fraction is shown in Fig. 5. For xB nearxdisB , f0(x

B, ) for constant x

B , has a single minimum at = 0corresponding to the disordered matrix phase. For xB nearxorB , f0(x

B, ) has two degenerate minima at = eq(x

B),corresponding to the two variants of the ordered phase. eq is afunction of the composition.

Minimization of the homogeneous free energy densityf0(xB, ) with respect to the order parameter yields thecomposition dependence of the free energy of a homogeneoussystem (see the bold curve in Fig. 6a). f0(xB, = 0)corresponds to the free energy of the disordered phase. Thefree energy of the ordered phase must be calculated byreplacing in the Landau polynomial by its equilibriumvalue eq(xB). eq(xB) is obtained by minimizing the freeenergy density for fixed compositions xB (see Fig. 6b). 1 fis the driving force for precipitation of the ordered phase ina supersaturated disordered structure with composition xB,0.At the diffuse interfacial regions between an ordered and adisordered domain, the concentration and the order parameterare forced to vary continuously between their equilibriumvalues. Therefore, f0 is not minimal at the interfaces. Theexact evolution of f0 across an interface depends on thegradient energy coefficients associated with the gradients in thecomposition and the order parameter. The dotted line in Fig. 6shows a possible evolution of the homogeneous free energydensity across a precipitatematrix interface. 1 f0 indicates thelocal contribution to the interfacial energy from the diffuseinterfaces.

The phenomenological parameters A, B,C, D must be cho-sen so that the composition dependence of the equilibrium freeenergy f0(xB, (xB)) agrees with experimental information.Most important is that the driving force for precipitation and theFig. 6. (a) Minimization of the Landau free energy density (18) with respectto the order parameter yields the composition dependence of the free energydensity for a homogeneous system (bold line). 1 f is the driving force forprecipitation of the ordered phase in a supersaturated disordered structure withcomposition xB,0. The dotted line gives a possible evolution of f0 acrossan interfacial region between a precipitate and the matrix. 1 f0 indicates thecontribution of the homogeneous free energy density to the interfacial energyin a heterogeneous system. (b) Landau free energy (18) as a function of theorder parameter at fixed compositions xB = 0.6 and xB = 0.8.

equilibrium composition of the ordered and disordered phaseare reproduced. The exact dependence of the non-equilibriumfree energy f0(xB, ) on the order parameter is not critical aslong as the number of minima and the symmetry properties arecorrectly reproduced.3

In the case of the ordered precipitates with L12 structure infcc Ni-based superalloys, the Landau free energy polynomialis a function of three order parameters and the composition

276 N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294[24,25,76,26,108]. The coefficients are chosen so that,for compositions near that of the ordered phase, it hasfour degenerate minima at (1, 2, 3) = (1, 1, 1)eq,(1,1, 1)eq, (1, 1,1)eq and (1,1,1)eq, and forcompositions near that of the disordered phase, it has a singleminimum at (1, 2, 3) = (0, 0, 0).

5.1.2. Interfacial energy

For the anti-phase domain structure in Fig. 3 represented bya single order parameter field, the interfacial energy and widthcan be calculated analytically [17,74]. In this specific situation,the total free energy of the system has the form

F = Fbulk + Fint =V

[f0()+

2( )2

]dr (19)

with f0 given by relation (16). The specific interfacial energyint (J/m3) is related to the gradient energy coefficient and thedepth of the minima of the homogeneous free energy (1 f0)maxas

int =42

3

(1 f0)max (20)

and the interfacial thickness as

l = 2

(1 f0)max. (21)

The value of depends on the definition of the thickness of adiffuse interface, as strictly speaking a diffuse interface reachesinfinity. For systems with more than one phase-field variable,the interfacial energy must be calculated numerically [26]. Itis generally true that the interfacial energy and the width ofthe profile of the phase-field variables at interfaces increase forlarger values of the gradient energy coefficients k and . Ifthe set of phase-field variables contains composition fields, onlythe gradient energy coefficients can be adjusted to fit both theinterfacial energy and the width, since f0 is determined by thecomposition dependence of the chemical free energies of thecoexisting phases (see Fig. 6).

Anisotropy in interfacial energy is usually introduced intothe free energy through the gradient terms of the orderparameter fields, for instance [109,24],

F =V

[f0(xB, k)+

12

3i, j=1

i jxBri

xBr j

+12

pk,l=1

3i, j=1

i jklk

ri

l

r j

]dr (22)

for ordering in fcc alloys (fcc to L12 structure). The expressionwas derived from the microscopic theory of Khachaturyan [24].i j is a second-rank tensor. The cross terms with k and l allowus to distinguish different types of interface depending on theorientation of the neighboring domains [109]. In [110], a freeenergy expansion with fourth-rank gradient terms was proposedfor systems with cubic anisotropy in interfacial energy.Fig. 7. Successive steps by which a coherent multiphase structure is formed andthe associated elastic strain is introduced into the material in micro-elasticitytheory (according to Khachaturyan [23]). (a) First the volume elements thatwill undergo a phase transformation are taken out of the stress-free material.(b) Then, these volume elements are allowed to transform to the new structureat the given temperature and pressure. They will adapt their equilibrium latticeparameters, volume and shape. The strains with respect to the original structure0i j are called the stress-free or transformation strains. (c) The transformedvolume elements must then be deformed so that they fit again in the system:toti j =

eli j +

0i j = 0. After this deformation, elastic strains (

eli j =

0i j ) and

stresses are present in the transformed material, since its dimensions differ fromthe equilibrium dimensions. (d) When the transformed material is put back intothe system, it can relax to a certain extent: eli j 6=

0i j . As a consequence,

elastic strains and stresses develop in the surrounding material.

5.1.3. Elastic misfit energy

Solid-state phase transformations usually induce elasticstresses into the material that may have a dominant effecton the microstructural evolution [111,112]. The size ofdomains and precipitates results from the competition betweeninterfacial energy and long-range elastic stresses. In thecase of isostructural phase transformations, ordering reactionsor symmetry reducing phase transformations, the producedmicrostructures are, at least in the initial stages of thetransformation, coherent, i.e. the lattice planes are continuousacross the interfaces. The lattice mismatch at interfaces, causedby a difference in crystal lattice parameters, is accommodatedby elastic displacements in the surrounding material. Themicro-elasticity theory for multiphase alloys of Khachaturyan[23,113,114] is applied to express the free energy contributiondue to coherency strains as a function of the phase-fieldvariables. Fig. 7 illustrates schematically how stress-free ortransformation strains and elastic stresses in coherent structuresare treated in micro-elasticity theory.

To relate the transformation strains 0i j (r ) to the local

values of the phase-field variables, a linear dependence on thecomposition fields and a quadratic dependence on the orderparameter fields are mostly assumed:

0i j (r ) = xBi j xB(

r )+k

ki j

2k (r ). (23)

xB(r , t) = xB(r , t) xB,0, with xB,0 a referencecomposition, which is for example taken equal to the overallcomposition of the alloy. xBi j are the lattice expansioncoefficients related to the composition. Following Vegards law,

xBi j (r ) =

1a

dadxB

i j (24)

N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294 277with a the stress-free lattice parameter. The ki j reflect thelattice mismatch between the product and parent phase. Forexample, for the cubic to tetragonal transformation shown inFig. 4 and using a coordinate system with axes parallel to theaxes of the cubic crystal, they have the form [27]

1 =

3 0 00 1 00 0 1

2 =1 0 00 3 00 0 1

3 =

1 0 00 1 00 0 3

.The stress-free strains 1 and 3 are related to the crystal latticeparameters ac, at and ct of the cubic and tetragonal structure intheir stress-free state

1 =at acac2eq

, and 3 =ct acac2eq

. (25)

The cubic structure is taken as the reference structure.The elastic strain eli j is the difference between the total strain

toti j and the stress-free strain 0i j , both measured with respect to

the same reference lattice,

eli j (r ) = toti j (

r ) 0i j (r )

= toti j (r ) xBi j xB(

r )k

ki j

2k (r ). (26)

The local elastic stress is then calculated by applying Hookeslaw from linear elasticity theory,

i j (r ) = Ci jkl(r )

elkl(r )

= Ci jkl(r )[totkl (r ) 0kl(

r )], (27)

where the Einstein summation convention is applied andCi jkl(r ) are the elements of the elastic modulus tensor. Sincecoexisting phases have usually different elastic properties, theelastic modulus tensor depends on the phase-field variables.Next, it is assumed that mechanical equilibrium is reachedmuch faster than chemical equilibrium, so that the equation formechanical equilibrium

i j

r j= 0 (28)

is valid. Solution of Eq. (28) in combination with relation (27),along with the appropriate mechanical boundary conditions,results in the equilibrium elastic strain field that is present inthe material after step 4 in Fig. 7.

To simplify the solution of the elasticity equation (28), thetotal strain toti j (

r ) is considered as the sum of a homogeneousstrain i j related to the macroscopic deformation of the system,and the local heterogeneous strain i j (r ) [23]:

toti j (r ) = i j + i j (r ). (29)

This treatment is only valid when the heterogeneities inthe microstructure are on a much smaller scale than themacroscopic size of the system. The homogeneous strain isdefined so thatVi j (

r )dr = 0. (30)

The heterogeneous strains are related to the local displacementswith respect to the homogeneously deformed material u (r )as

kl(r ) =

12

[uk(r )

rl+ul(r )

rk

], (31)

where ui denotes the i th component of the displacementvector u . Efficient methods have been developed to solvethe elasticity equations [4,66,115,116] for materials withheterogeneous elastic properties (Ci jkl(r)).

Once the elastic strains are obtained, the total elastic energyFel of the system is calculated through the usual expression,

Fel =12

VCi jkl(r )

eli j

elkldr (32)

for a system under an applied strain ai j = i j . In the presenceof an externally applied stress ai j (

r ), the elastic energycontribution should include the external work [25],

Fel =12

VCi jkl(r )

eli j

elkldr

V ai j (

r )i j (r )dr . (33)

The homogeneous strain is then obtained by minimizing theelastic strain energy with respect to the homogeneous strain.

The phase-field micro-elasticity model was applied toa variety of mesoscale phenomena in solid materials. Theinfluence of the elastic properties of the coexisting phaseson the morphological evolution in spinodal decompositionand further coarsening of the structure were studied manytimes [65,34,32,33,21,66,117]. The model also allows usto predict the shape and size of coherent precipitates, aswell as the splitting and coalescence of precipitates underinfluence of an applied stress [77,118,24,25,108,4,76]. Formartensitic transformations [27,28,119] and other structuraltransformations [35,79], the effect of the coherency stresses onthe shape and spatial distribution of the different orientationvariants were analyzed, as well as the influence of externallyapplied stresses on the structure. Furthermore, a phase-fieldmicro-elasticity model for the evolution of structures with voidsand cracks under an applied stress was proposed [7]. The micro-elasticity formalism was also extended towards polycrystallinematerials containing multiple grains with a different orientation[119121].

Although dislocations are crystal defects on the atomic scale,the stresses they cause on the mesoscale and their effect onmicrostructural evolution can be modeled using the phase-fieldmicro-elasticity approach. It was possible to reproduce solutesegregation (formation of Cottrell atmospheres) and nucleationand growth of precipitates around dislocation cores [4,122].Furthermore, the interaction between a moving dislocationloop and an array of precipitates [5] was simulated. Themovement and multiplication and annihilation of dislocationsunder an applied stress [6,123] and the formation of dislocation

278 N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294networks and substructures [124] were also studied by phase-field simulations. Up to now, most micro-elasticity modelsinvolving dislocations were for fcc crystals.

When transformation strains are involved, the molar volumeis not constant. Nevertheless, Eq. (12) for the chemical freeenergy is still valid, if considered in a number fixed frame4 witha constant reference molar volume V refm , i.e.

F(xB, k) =V ref

[f0,m(xB, k)

V refm+

2( xB)

2

+

k

k

2( k)

2

]dr , (34)

where f0 = f0,m/V refm and Vref

= V refm ntot, with ntot thetotal number of moles. V refm is the molar volume of a referencestructure, which must be the same as the structure referred toin Eq. (23) and Eqs. (25) for the formulation of the stress-freestrains. f0,m is the homogeneous free energy, expressed pernumber of moles, assuming the equilibrium lattice parametersat the given temperature and pressure. The free energies f0considered up to here are, in fact, of the form f0 = f0,m/V refm .

An approach similar to that of the elastic strain energyis applied to calculate the electrostatic or magnetic energycontribution when charged species or electric or magneticdipoles are involved. It is assumed that electric and magneticequilibrium are much faster than chemical equilibrium.Therefore, an electromagnetic equilibrium equation is firstsolved to obtain the electric or magnetic field for thecurrent distribution of charges and dipoles and then the totalelectrostatic or magnetic energy is expressed as a functionalof the phase-field variables. To characterize ferroelectric orferromagnetic domain growth, the order parameter fields aretaken equal to the coordinates of the polarization vector forferroelectric materials [125,29,30] or the coordinates of themagnetization vector for ferromagnetic materials [31].

Recently, elastoplastic phase-field models were proposed byGuo et al. [126] and Hu et al. [127]. Additional phase-fieldvariables are used to describe the plastic strain fields on themesoscale. The local strain tensor toti j then consists of an elastic

part eli j , a contribution due to lattice mismatches 0i j , and a

plastic part pli j . Moreover, besides the general terms in the freeenergy, a distortion strain energy is formulated as a function ofthe additional phase-field variables for the plastic strain.

5.2. Single-phase-field model for solidification

In the more phenomenologically oriented phase-fieldmodels, such as those for solidification, a phase-field is usedto distinguish two coexisting phases. The following functionalis an example of a free energy expression used in simulations4 This involves that the spatial coordinates expressed in a reference framerelated to the physical dimensions of the system are transformed to a referenceframe that is defined so that the number of atoms per unit volume of the newreference frame is constant.for isothermal solidification of a single phase from a binaryliquid: [46]

F =V

[f0(xB, , T

)+

2( xB)

2+

2( )2

]dr , (35)

with f0(xB, , T ) the homogeneous free energy density at agiven temperature. In many solidification models, the gradientin composition is not considered, i.e. = 0. The molar volumeis usually assumed to be constant.

5.2.1. Homogeneous free energy densityIn the single-phase-field model, the homogeneous free en-

ergy density consists of an interpolation function f p(xB, , T )and a double-well function wg():

f0(xB, , T) = fp(xB, , T

)+ wg(). (36)

The double-well potential g()

g() = 2(1 )2 (37)

has minima at 0 and 1 and w is the depth of the energywells. w is either a constant [49] or depends linearly on themolar fraction, namely w = (1 xB)wA + xBwB [44]. Theinterpolation function fp combines the free energy expressionsof the coexisting phases f (xB, T ) and f (xB, T ) into onefree energy expression, which is a function of the field variablesxB and , using a weight function p():

fp(xB, , T) = (1 p()) f (xB, T )

+ p() f (xB, T). (38)

In this expression, it is assumed that equals 1 in the-phase and 0 in the -phase. The functions f (xB, T )and f (xB, T ) are homogeneous free energy expressionsfor the composition and temperature dependence of the-phase and the -phase (see Fig. 8). They are usuallyconstructed, using a regular substitutional solution model ortaken from a thermodynamic CALPHAD (CALculation ofPHAse Diagrams) description of the phase diagram.

p() is a smooth function that equals 1 for = 1 and 0 for = 0 and has local extrema at = 0 and = 1. Mostly, thefollowing function [42]

p() = 3(62 15 + 10) (39)

is used, for which p() = 30g(). The functions g() andp() are plotted in Fig. 9.

Fig. 10 gives a three-dimensional view of f0 as a functionof xB and , obtained from relations (36) and (38) with w = 1and for the free energies of the -phase and -phase plottedin Fig. 8. Minimization of f0 with respect to at a fixedcomposition xB , gives f

(xB, T) and = 0 or f (xB, T

)

and = 1, depending on which is the lower. The minimacorrespond with the lower envelope in the projection of f0 onthe xB f0-plane shown in Fig. 11a. Thus, in a two-phasestructure, equals 0 in the -domains and 1 in the -domainsand the free energy equals, respectively, f (xB) and f

(xB).However, at the diffuse interfaces, is forced to take values in

N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294 279Fig. 8. Phenomenological free energy densities of the phases and as afunction of composition xB and at temperature T . According to Gibbsianthermodynamics, the system decomposes into two phases for systems with a

mean composition xB,0 between xB and x

B , namely with composition x

B

and with composition xB . The equilibrium compositions of the coexistingphases are given by the tangent rule. The dotted line indicates the evolution ofthe interpolation function across a diffuse interface.

Fig. 9. Double-well potential g() and weight function p() used to constructthe homogeneous free energy density f0 for a two-phase system.

Fig. 10. Three-dimensional graphic of the homogeneous free energy f0 of atwo-phase system according to Eq. (36).Fig. 11. (a) Projection of f0 on the xB f0-plane. For homogeneous systems,minimization of the interpolation function f p with respect to givesf p,0(xB , T

) = min( f (xB , T ), f (xB , T )). The lower envelope gives forfixed values of xB the minimum of f0 with respect to . (b) Projection of f0 onthe f0-plane. The lower envelope gives for fixed values of , the minimumof f0 with respect to xB .

between 0 and 1. Then g() 6= 0 and the interpolation functionf p(xB, , T

) deviates respectively from f (xB, , T) and

f (xB, , T). The dotted line in Fig. 8 indicates how f p may

evolve across a diffuse interface between the -phase and the-phase. When = 0 in the free energy density f (35), theevolution of f0 across a diffuse interface is given by the minimaof f0 with respect to xB for fixed values of . It correspondsto the lower envelope in a projection of f0 on the f0-plane,which is shown in Fig. 11b.

5.2.2. Interfacial energyIn general, the gradient terms, the double-well function

wg() and the interpolation function f p(xB, ) contributeto the interfacial energy. The contribution from the double-well potential is w1g, with 1g as indicated in Fig. 9. Thecontribution from the interpolation function 1 f p is given bythe difference between the dotted line and the common tangentline in Fig. 8. In addition to the gradient energy coefficients,the double-well coefficient w can be modified to adjust theinterfacial energy and width. The interfacial energy and widthare larger for larger values of the gradient energy coefficients.The interfacial energy increases and the interfacial thicknessdecreases with the depth of the double-well potential, whichis proportional to w. However, there is no analytical relationbetween the parameters in the phase-field model and theinterfacial energy and width [46]. Moreover, the width of the

280 N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294profile of the concentration variable may differ from that of thephase-field.

In the model of Tiaden et al. [59] and Kim et al. [49], thegradient in concentration is not included in the free energydensity, the parameterw in front of the double-well is a constantand an extra restriction is imposed on the local concentration atthe interfaces, namely

xB = (1 p())xB + p()xB (40)

and

f0xB

(xB) = f0xB

(xB). (41)

They consider the interface as a mixture of two phases withdifferent composition (phase with composition xB and phase

with composition xB) and impose that the chemical potential

in both phases must be the same. xB and xB may differ from the

equilibrium bulk compositions xB and xB indicated in Fig. 8.

As a consequence, the interpolation function f p follows thecommon tangent line at the interfaces and consequently doesnot contribute to the interfacial energy. Then, the interfacialenergy and thickness only depend on the gradient term in andthe double-well potential. They can be calculated analytically,giving

int =

w

32

(42)

and

l = 2

w, (43)

where depends on the definition of the interfacial thickness.The advantage of such a model is that the parameters w and in the phase-field model are in a straightforward way related tophysical properties int and l.

5.2.3. Anisotropy

The interfacial energy of a solidliquid interface may dependon the orientation of the crystal with respect to the interface.Also in the case of solidsolid phase transformations, thereare usually low-energy interface orientations resulting in, forexample, plate-like particles. To obtain realistic simulations,this anisotropy is included into the phase-field model by giving the same orientation dependence as the interfacial energy,

as the interfacial energy is proportional to .

For two-dimensional systems, the normal to the interfacen = (n1, n2), which is in fact the normal to contours ofconstant value of a phase-field variable , is given by

(n1, n2) =1(

r1

)2+

(r2

)2(

r1,

r2

), (44)and the angle between the normal to the interface and ther1-axis by

tan() =n2n1

=/r2/r1

. (45)

For weak anisotropy, the following formulation is commonlyapplied for two-dimensional systems [45]: = (1+ cos(k)), (46)

with < 1/15 a measure of the strength of anisotropy and k theorder of anisotropy. For example for cubic anisotropy, k equals4. According to the GibbsThomson condition +d2/d > 0(with the interfacial energy and the orientation of thenormal to the interface), in Eq. (46) must be smaller than1/15.

In order to describe strong anisotropy in interfacial energy,resulting in faceted morphologies, functions with deep cuspsat the low-energy orientations are required. Examples of suchanisotropy functions, which were used to distinguish the largedifference in energy between a coherent and an incoherentinterface, are = (1+ (| sin()| + | cos()|)) , (47)

resulting in square particles [128], and

=

1+ (1+ | cos()|) , (48)

resulting in plate-like particles [129,98,130]. The equationsmust be regularized as described in [128,98] to circumvent thediscontinuities in the first derivative. In both cases, definesthe amplitude of the anisotropy. More functions for

(n ) or

() have been proposed to simulate the growth of faceted

crystals [131133].A similar approach is used to describe anisotropy in three

dimensions [48,134,135], giving for example =

[1+

[cos4(1)+ sin4(1)

(1 2 sin2(2) cos2(2)

)]](49)

for weak cubic anisotropy, where 1 and 2 define theorientation of the normal to the interface in three dimensions.

If the energy gradient coefficient depends on theorientation of the interface, the interfacial thickness variesalong the interface; see for example Eq. (43). In the case ofstrong anisotropy, these variations in interfacial thickness mayresult in artifacts, such as enhanced particle coalescence alonghigh-energy interfaces or numerical pinning along low-energyorientations [136]. Therefore, a better approach would be tomake both the energy gradient coefficient and the height ofthe double-well potential orientation dependent in such a waythat the orientation dependence of the interfacial energy isreproduced, while the interfacial thickness is constant along theinterface.

5.2.4. Non-isothermal solidificationFor non-isothermal solidification, a thermodynamic entropy

functional should be considered instead of a free energy

N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294 281functional [41,137,42,138], since the temperature is notprescribed. For example,

S =V

[s0(e0, xB, )

2( xB)

2

2( )2

]dr , (50)

where the homogeneous entropy density s0 depends on theinternal energy density e0, the phase-field and the composition.The internal energy is a function of temperature. The internalenergy e0, Helmholtz energy f0 and entropy s0 are related ase0 = f0 + s0T , with T the temperature. At thermodynamicequilibrium the entropy functional S is maximal.

5.3. Multiphase-field model

Multiphase-field models have been developed for thesimulation of phase transformations that involve more thantwo phases, such as eutectic and peritectic solidification. Themicrostructure is described by a vector of multiple phase-fields k , which represent the local fractions of the differentphases. The free energy as a functional of the molar fractionfields xi (r , t) and phase-fields k(r , t) at a fixed temperatureT usually has the form [58,100,139]

F(xi , , T) =

V

[f p(xi , , T

)+ g()

+

pj

282 N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 2682946.1. Time-dependent GinzburgLandau equation

The non-conserved variables k(r , t) and k(r , t) governa time-dependent GinzburgLandau equation, also referredto as a AllenCahn equation. For the order parameter fieldrepresentation,

k(r , t)

t= Lk

F

k(r , t)

= Lk

[ f0k

k

k

](59)

is obtained for k = 1 . . . p, where F and f0 are functions ofthe conserved and non-conserved field variables as discussedin Section 5.1. Lk and k may depend on the local valuesof the field variables to introduce anisotropy or compositiondependence of the interfacial energy or mobility. In the single-phase-field model, for example in the case of a systemrepresented by a single phase-field variable and a compositionfield xB , the evolution of is given by

(r , t)

t= L

F(xB, )

(r , t)

= L

[ f0(xB, )

()

], (60)

where it is assumed that may depend on the phase-field. Inthe multiphase-field model, the technique of the -multiplier isapplied to account for restriction (10) on every position in thesystem:

k(r , t)

t= L

F (xB, l)

k(r , t)

, (61)

with

F =V

[f (xi , l , T

)+

(p

l=1

l 1

)]dV

= F +V

[

(p

l=1

l 1

)]dV (62)

and F and f (xi , l , T ) respectively the total free energy andthe free energy density, which are discussed in Section 5.3.Elimination of gives

k(r , t)

t=

L

p

pl=1

(F

k(r , t)

F

l(r , t)

), (63)

with p the number of phase-fields.The L and Lk in Eqs. (59), (60) and (63) are positive

kinetic parameters, related to the interfacial mobility . It wascalculated by Allen and Cahn [74], assuming that l/ 1with 1/ the local curvature of the interface and l the interfacialthickness, that the interfacial velocity equals

v = L

(1

)(64)

for curvature driven coarsening of the anti-phase domainstructure in Fig. 3. In classical sharp-interface theories forcurvature driven motion of interfaces, the velocity of aninterface is given by

v = int(1/) = (1/), (65)

where is called mobility and reduced mobility.Accordingly, the product L in the phase-field model is equal tothe reduced mobility in classical models for interfacial motion,which is a measurable quantity [146]. Notice that, for curvaturedriven grain boundary movement, the velocity of the interfaceis not affected by the form of the non-equilibrium free energydensity f0() (see e.g. Eq. (16)).

Analogous to the anisotropy in interfacial energy (Sec-tion 5.2.3), anisotropy in interfacial mobility is introduced bygiving L a dependence on the interfacial orientation throughthe angle defined by relation (45) [133].

6.2. CahnHilliard equation

The evolution of conservative phase-field variables, suchas the molar fraction field xB(r , t), obeys a CahnHilliardequation, for example,

1Vm

xB(r , t)

t= M

F(xB, k)

xB(r , t)

= M

[ f0(xB, k)

xB

xB(r , t)

]. (66)

The kinetic parameter M can be a function of the orderparameter fields and the composition. The CahnHilliardequation is essentially a diffusion equation of the form

1Vm

xB(r , t)

t=

J B, (67)

where the diffusion fluxJB is given by

J B = M

F

xB

= M

[ f0(xB, k)

xB

xB(r , t)

]. (68)

Mass conservation is accordingly ensured, as well as the fluxbalance at the interfaces (compare with Eq. (3) in sharp-interface models). The diffusion fluxes are defined in a numberfixed reference frame (the number of moles per unit volume inthe number fixed reference frame, is constant in time) and theparameter M relates to the interdiffusion coefficient D as

M =VmD

2Gm/x2B, (69)

which gives for an ideal solution

M =DxBVmRT

. (70)

M can also be expressed as a function of the atomic mobilitiesof the constituting elements MA and MB [147],

M =

(1Vm

)xB(1 xB)[xBMA + (1 xB)MB]. (71)

N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294 283Atomic mobilities are related to tracer diffusion coefficients.If sufficient experimental information is available, expressionsfor the temperature and composition dependence of the atomicmobilities Mi can be determined using the DICTRA (DIffusionControlled Transformation) software (see Section 7.1.1). Toaccount for differences in diffusion properties between thecoexisting phases and for enhanced grain boundary diffusion,M may depend on the order parameter fields or phase-fields.

6.3. Heat flow and other transport phenomena

Besides diffusion, other transport processes can have adominant effect on the microstructure. Using the formalismof linear non-equilibrium thermodynamics [145], it isstraightforward to extend the standard phase-field model toinclude phenomena such as heat diffusion, convection andelectric current.

6.3.1. Non-isothermal solidificationFor non-isothermal solidification [41,137,42,138], the

following equations were obtained for an entropy functional Sof the form (50):

e

t=

MT

s

e

= MT

1T, (72)

t= L

[s

+ 2

]= L

[1T

f

2

](73)

and

1Vm

xBt

= M

[s

xB+ 2xB

]= M

[1T

f

xB 2xB

]. (74)

The kinetic parameter MT in the heat equation (72) can berelated to the thermal diffusivity. For a constant temperature,Eqs. (73) and (74) are equivalent with the GinzburgLandauequation (60) and the CahnHilliard equation (66).

6.3.2. ConvectionCompared to purely diffusive transport, convection in the

liquid phase enhances the transport of heat and mass duringsolidification. One of the methodologies to include convectionis to combine the phase-field equations with a NavierStokesequation where the viscosity depends on the phase-field [57,148151]. In this approach, both the solid and liquid phase aretreated as a Newtonian viscous fluid with the viscosity of thesolid phase several orders of magnitude larger than that of theliquid phase. A full thermodynamic derivation of the equationsfor mass, energy and momentum conservation and an equationfor the evolution of the non-conserved phase-field was givenby Anderson et al. [152]. Very recently the phase-field methodwas also coupled with the lattice Boltzmann method [153], analternative technique for simulating fluid flow.6.3.3. Electric currentBrush [154] derived a phase-field model that considers the

effect of heat flow and an applied electric current on crystalgrowth. The model reproduces typical phenomena related tothe combination of heat flow and electric current, such as Jouleheating, the Thompson effect and the Peltier effect.

6.3.4. Plastic deformationIn phase-field models for elastoplastic deformation, the

evolution of the phase-field variables that describe the plasticstrain fields is given by time-dependent GinzburgLandauequations with the free energy functional equal to the distortionstrain energy [126] or given by an objective function basedon a yield criterion [127]. In both cases, the kinetic equationsfor plastic flow are only defined in regions where the effectivestress exceeds a critical value given by the yield criterion.

6.4. Stochastic noise term

Stochastic Langevin forces are sometimes added to thephase-field equations to account for the effect of thermalfluctuations on microstructure evolution, for example,

k(r , t)

t= L

F(xB, j )

k(r , t)

+ k(r , t) (75)

1Vm

xB(r , t)

t= M

F(xB, k)

xB(r , t)+ B(

r , t), (76)

with k(r , t) non-conserved and B(r , t) conservedGaussian noise fields that satisfy the fluctuationdissipationtheorem [18,19]. The local magnitudes of the Langevin noiseterms must be scaled appropriately so that the noise in thebulk phases and interfaces in the phase-field simulations canbe related quantitatively with temperature fluctuations that arepresent in real experiments [155].

Mostly, the Langevin terms are used purely to introducenoise at the start of a simulation and are switched off after a fewtime steps. For example, to initiate precipitation of an orderedphase in a metastable supersaturated disordered matrix or forthe onset of dendritic growth in solidification. Granasy et al.[156,157] employed the Langevin noise terms to simulate thenucleation of crystal seeds in undercooled melts.

7. Quantitative phase-field simulations for alloy develop-ment

The early phase-field simulations showed that the phase-field technique is a general and powerful technique forsimulating the evolution of complex morphologies. Simulationscould give important insights into the role of specificmaterial or process parameters on the pattern formationin solidification and the shape and spatial distribution ofprecipitates or different orientation domains in solid-state phasetransformations. However, the results were rather qualitativeand there are two major difficulties that complicate quantitativesimulations for real alloys. The phase-field equations containmany phenomenological parameters, which are often difficult

284 N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294

5 Department of Materials Science and Engineering, KTH, Stockholm.6 CompuTherm LLC.7 Department of Materials Science and Engineering, KTH, Stockholm.8 ACCESS, Aachen.to determine for real alloys. Furthermore, because in realalloys the width of the interfaces is several orders smaller thanthe microstructural features (dendrites, grains, precipitates)and processes (heat and mass diffusion), massive computerresources are required to resolve the evolution of the phase-fieldvariables at the interfaces appropriately and, at the same time,cover a system with realistic dimensions. Often, simulationsare restricted to simplified or model systems and results areaffected by artificial boundary effects. To make the phase-field technique attractive for technical applications, in currentresearch much attention is given to the quantitative aspectsof the simulations. Different methodologies are applied todetermine the parameters and to deal with the inconveniences ofthe diffuse interfaces. Moreover, advanced numerical solutiontechniques are proposed and powerful parallel codes aredeveloped.

7.1. Parameter determination

As is the case for all mesoscale and macroscalemodeling techniques, the phase-field equations contain a largenumber of parameters that must be determined in orderto introduce the material specific properties for real alloys.The parameters in the homogeneous free energy densitydetermine the equilibrium composition of the bulk domains,the gradient energy coefficients and, if applicable, the double-well coefficient are related to the interfacial energy andwidth, the kinetic parameter in the CahnHilliard equation isrelated to diffusion properties and the kinetic parameter in theGinzburgLandau equation to the mobility of the interfaces.For solid-state phase transformations, the microelasticitytheory requires also the elastic moduli of all coexistingphases. The parameters may show directional, compositionaland temperature dependence. In particular, the orientationdependence of the properties is decisive for the morphologicalevolution. Due to the large number of parameters and alsobecause some properties or dependences are difficult todetermine experimentally, parameter assessment for phase-field models is not straightforward. To reduce the dependenceon experimental input, the phase-field technique has beencombined with the CALPHAD approach for multicomponentalloys and with atomistic calculations.

7.1.1. CALPHAD approach for multicomponent alloysThe CALPHAD (CALculation of PHAse Diagrams) method

was originally developed for the calculation of phase diagramsof multicomponent alloys using thermodynamic Gibbs energyexpressions. The required Gibbs energy expressions of thedifferent phases are determined by fitting the parametersin a thermodynamic model using experimental informationon thermodynamic properties and phase equilibria. TheCALPHAD approach provides an efficient and consistenttool for constructing temperature-dependent and composition-dependent Gibbs energy expressions for multicomponentsystems. Different thermodynamic solution models weredeveloped to describe the temperature and compositiondependences of the free energy, such as the substitutionalsolution model (atoms can freely mix over all lattice sites) andthe sublattice model (atoms are only allowed to mix withindistinct sublattices). In the CALPHAD method, expressions forthe Gibbs energy for multicomponent systems are constructedthrough an extrapolation from the binary subsystems. Thismeans that the expressions of the binary systems are combinedaccording to a mathematical model, resulting in a descriptionof the multicomponent system. Deviations from the modelare treated using higher-order interactions parameters. Whenthe parameters of the binary system are known, only thehigher-order interaction terms must be determined accordinglyto obtain the Gibbs energy of the multicomponent system.Thanks to the standardized approach, the work of differentresearchers can be combined in large-scale internationalprojects for obtaining thermodynamic databases for complexmulticomponent alloys. For many industrial alloy systems,such as steels, light (Al, Ti, Mg) alloys, soldering alloys,ceramic materials and slag systems, thermodynamic databasescontaining expressions for the interaction parameters areavailable. Furthermore, computer programs, such as Thermo-Calc5 and Pandat,6 were developed for the calculation of thephase diagrams and optimization of the parameters in the Gibbsenergy expressions.

Within the DICTRA (DIffusion Controlled TRAnsforma-tions) software7 for the simulation of diffusion controlled phasetransformations for multicomponent alloys (based on a sharp-interface model), temperature-dependent and composition-dependent expressions for the atomic mobilities (see Eq. (71))are obtained in a similar way as the Gibbs energy expressionsin the CALPHAD method [147,158]. The parameters are de-termined using experimentally measured tracer, interdiffusion,and intrinsic diffusion coefficients. Unfortunately, the numberof assessed systems in the DICTRA database is rather limited.

In the single-phase-field and multiphase-field models, wherethe free energy density is constructed as an interpolationfunction of the free energies of the coexisting phases, theGibbs energy expressions from CALPHAD databases canbe used directly in the phase-field free energy. The phase-field method was for example combined with thermodynamicCALPHAD databases and kinetic DICTRA databases forsimulating solidification of ternary AlMgSi-alloys [63] andcommercial MgAlZn and AlSiCuMgNi alloys [159],the growth of perlite [106], the morphological evolutionin AlNiCr diffusion couples [70,160], Ostwald ripeningin AlNiCr alloys [161], diffusion controlled precipitategrowth and dissolution in TiAlV [162] and the growthand coarsening of the intermetallic compound layer in CuSnleadfree solder joints [163,164]. More details on the couplingof phase-field simulations with thermodynamic and phasediagram calculations are explained in [139]. A phase-fieldsimulation software MICRESS8 has been developed based on

N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294 285the multiphase-field model, which is coupled with the Thermo-Calc software for the thermodynamic calculations.

In the case of the Landau free energy expressions, it ismore complicated to determine the parameters in the Landaufree energy f (xi , eq(xi )) from the CALPHAD expressionsof the coexisting phases. Only for ordering reactions can theCALPHAD free energy expression be adopted directly in theLandau polynomial used in the phase-field model, since inthe CALPHAD approach the ordered and disordered phaseare described with a single free energy expression that iscoincidentally very similar to the Landau free energy density(18) used in the phase-field model for ordering reactions. Thiswas illustrated by Zhu et al. [165] for quantitative simulationsof the growth of ordered -precipitates in Ni-based alloys.

Through extrapolation from lower-order systems, theCALPHAD approach allows us to obtain a full thermodynamicdescription of a multicomponent system using a limited amountof experimental information. From these thermodynamicdescriptions, phase relationships and thermodynamic propertiescan be calculated for experimentally uninvestigated regions.The CALPHAD methodology is successful for determiningbulk free energy expressions, since for most systems oftechnical interest there is a large amount of experimentaldata on thermodynamic properties and phase relationshipsavailable. In the case of the DICTRA mobility database,only a few systems have been optimized because therequired experimental information is not available. Onlya few laboratories are specialized in measuring diffusioncoefficients, whereas the determination of phase equilibria andthermodynamic properties is much more common. Moreover,more mobility expressions must be optimized (a temperature-dependent and composition-dependent mobility expressionmust be determined for each component in each phase insteadof a single expression for each phase for the Gibbs energy).Application of the CALPHAD approach for modeling theinterfacial energy and mobility in multicomponent systemswould be interesting, for example by optimizing temperature-dependent and composition-dependent expressions for theparameters in a model that describes the orientation dependenceof the interfacial properties. However, it is even more difficultto measure interfacial energy and mobilities than to measurediffusion properties. Experimental information on interfacialproperties is scarce, and relates mostly to pure materials.

7.1.2. Coupling with atomistic calculationsThe phase-field simulation technique was also combined

with ab initio calculations and other atomistic simulation tech-niques to obtain difficult-to-measure parameters. Vaithyanathanet al. [26,166,167] presented a multiscale approach based onab initio calculations, mixed-cluster expansion Monte Carlosimulations and the phase-field method, for investigating thegrowth and coarsening of -Al2Cu precipitates during agingof an Al alloy. The free energy density, the interfacial energyand anisotropy, the stress-free lattice parameters and the elas-tic constants were determined based on ab initio calculations.Bishop and Carter [168] propose a coarse graining method forrelating the atomic structure of a grain boundary obtained fromatomistic simulations, with the evolution of a continuous orderparameter across a diffuse interface in the phase-field model.Hoyt, Asta and Karma [169,170] determined from atomisticmolecular dynamics simulations, the interfacial energy and mo-bility and their anisotropies for pure Ni, Au and Ag. The atom-istic calculations were coupled with a phase-field model [171]to study quantitatively the dendritic growth in these metals asa function of the undercooling. They also computed the liquiddiffusivities of pure Cu and Ni in the vicinity of their meltingpoints [172].

In principle, all parameters in the phase-field model canbe calculated using an atomistic technique. A multiscaleapproach that combines ab initio, other atomistic simulationtechniques such as molecular dynamics, cluster expansionand atomistic Monte Carlo, and the mesoscale phase-fieldtechnique, would result in a quantitative and fully predictivesimulation technique. However, at this moment, the quantitativeresults from atomistic simulations themselves are not veryreliable. There may be large deviations, up to 200% or300%, between values for the same properties calculated usingdifferent atomistic techniques or different approximations forthe interaction potential. Moreover, most atomistic simulationtechniques still require a number of model parameters that mustbe determined using experimental data.

7.2. Numerical solution of the phase-field equations

For the numerical solution of the phase-field equations, thecontinuous system is projected on a lattice of discrete points,as is illustrated in Fig. 12. The phase-field equations are alsodiscretized, resulting in a set of algebraic equations. Solutionof the algebraic equations yields the values of the phase-field variables in all lattice points. For the lattice depictedin Fig. 12, the set of algebraic equations contains N1 N2equations and the same number of unknowns for each phase-field variable. The lattice spacing 1x must be fine enoughto resolve the interfacial profile of the phase-field variables,i.e. there must be a few grid points within the interfacialregion. At the same time, the dimensions of the system l1 enl2 must be large enough to cover the processes occurring ona larger scale. As a consequence, the numerical solution ofthe phase-field simulations for actual alloy systems involvesan enormous number of lattice points, since N1 = l1/1x andN2 = l2/1x . Since, at each time step, the value of the phase-field variables must be computed for all the grid points, phase-field simulations are quite computationally intensive. Moreover,a smaller lattice spacing involves a smaller time step in order tomaintain numerical stability. Large computer memory is alsorequired to treat the huge algebraic systems of equations withmany unknowns.

A finite difference discretization technique using uniformlattice spacing, and with a central second-order stepping inspace and forward stepping in time, is most widely usedbecause of its simplicity. For the two-dimensional system inFig. 12, discretization of the GinzburgLandau equation then

286 N. Moelans et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 268294

Fpdfttd

g

Tatpfvtscasoeaastmitt

ma

ig. 12. Projection of a continuous phase-field variable on a lattice of discreteoints for a two-dimensional system at time step n. Discretization of the partialifferential phase-field equations results in a set of N1N2 algebraic equationsor each phase-field variable, that allows us to calculate, for each time step n,he values of the phase-field variables nr,s in all lattice points. l1 and l2 arehe dimensions of the system, N1 and N2 the number of grid points in eachimension and 1x is the lattice spacing.

ives the following set of algebraic equations:

n+1r,s nr,s

1t= L

[( f0

)nr,s

nr1,s +

nr,s1 4

nr,s +

nr+1,s +

nr,s+1

(1x)2

]. (77)

he values of the phase-field variables at time step n + 1re directly calculated from the values at time step n. Theechnique can be combined with Neumann, Dirichlet anderiodic boundary conditions. However, it is well known thator such an explicit time stepping, the time step 1t must beery small to keep the numerical solution stable. The maximumime step is orders of magnitude smaller than the typical timepan of microstructural evolution phenomena, resulting in longomputation times. Sometimes a backward time stepping ispplied to improve the stability conditions. The right-handide in Eq. (77) is then evaluated on time step n + 1 insteadf n, resulting in a system of coupled non-linear algebraicquations. The equations must be linearized, for example usingNewtonRaphson method, and ite