Phase-Field Model for Microstructure Evolution at the Mesoscopic Scale

MR43CH05-Steinbach ARI 22 February 2013 12:52

RE V I E W

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Phase-Field Modelfor Microstructure Evolutionat the Mesoscopic ScaleIngo SteinbachInterdisciplinary Centre for Advanced Materials Simulation, Ruhr University, Bochum 44780,Germany; email: [email protected]

Annu. Rev. Mater. Res. 2013. 43:5.1–5.19

The Annual Review of Materials Research is online atmatsci.annualreviews.org

This article’s doi:10.1146/annurev-matsci-071312-121703

Copyright c© 2013 by Annual Reviews.All rights reserved

Keywords

phase field, interfaces, diffusion, spinodal decomposition

Abstract

This review presents a phase-field model that is generally applicable to ho-mogeneous and heterogeneous systems at the mesoscopic scale. Reviewedfirst are general aspects about first- and second-order phase transitions thatneed to be considered to understand the theoretical background of a phasefield. The mesoscopic model equations are defined by a coarse-grainingprocedure from a microscopic model in the continuum limit on the atomicscale. Special emphasis is given to the question of how to separate the inter-face and bulk contributions to the generalized thermodynamic functional,which forms the basis of all phase-field models. Numerical aspects of thediscretization are discussed at the lower scale of applicability. The modelis applied to spinodal decomposition and ripening in Ag-Cu with realisticthermodynamic and kinetic data from a database.

5.1

Review in Advance first posted online on March 1, 2013. (Changes may still occur before final publication online and in print.)

Changes may still occur before final publication online and in print

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INTRODUCTION

A phase field in its original meaning denotes the region in the phase diagram of an alloy in whichone crystallographic phase is stable. This region of the phase diagram can be a wide compositionrange in the case of a solution phase or a very narrow region in the case of a compound phase.This region is characterized by the concentration of the alloying elements, temperature, andpressure. Within one single phase field, the corresponding crystallographic phase has a minimumGibbs energy compared with all the other phases of the same composition and compared withall the combinations of different phases with individual compositions that add up to the originalcomposition according to the lever rule. To identify such a phase field in composition space, theso-called CALPHAD [CALculation of PHAse Diagrams (1)] method can be used. This methodrelies on tabulated functions of the Gibbs energies of the individual phases, depending on thecomposition, temperature, and pressure, and uses a global minimization of the total Gibbs energy.Each phase is treated as an entity in thermodynamic equilibrium, i.e., with a volume large enoughto neglect surface effects. Outside of a single phase field, a piece of matter will split into differentphases whose compositions and fractions are determined by the requirements of global Gibbsenergy minimization and equality of chemical potentials.

The phase-field method in the context of this review leaves the limit of thermodynamicalequilibrium. It is designed to describe the evolution of a piece of matter, prepared in an arbitraryoff-equilibrium state, toward global equilibrium. The phase, in the original thermodynamic sense,is identified by its crystallography. Only the word field has a different definition. Instead of charac-terizing the region in concentration space, the term field in a phase-field model characterizes thelocation in space �x and time t in which a crystallographic analysis of matter would detect the phaseunder consideration. This discussion brings us to some important issues: resolution in space andtime, interfaces between microstructures and phases, and the occurrence of metastable states. I tryto clarify these issues in the following section. For now I simply note the common understandingof all phase fields: (a) A phase is characterized by its crystallography and composition. (b) In equi-librium, any piece of matter belonging to one crystallographic phase will have a composition insidethe single phase field of a phase diagram. The phase-field method, however, does not deal onlywith equilibrium states. Usually we nucleate a new phase within the single phase field of a differentphase, e.g., nucleate a solid in an undercooled liquid. Alternatively, we nucleate one new phasefrom another phase whose composition lies in the two-phase field between two stable phases. Thetwo-phase fields may be related to different crystallographic phases. Then there is a nucleationbarrier to be overcome. One speaks of heterogeneous phases or of the two phases, indexed by thephase fields, that belong to the same crystallographic phase but are distinct in their composition.The latter case corresponds to a miscibility gap in which one phase, e.g., a molten monotecticalloy, decomposes into two variants of the same phase, but with different compositions. In mostcases, this separation into variants or subphases is a first-order phase transition. Only at the criticalpoint, i.e., at the temperature and composition at which the two variants become identical, maywe describe the phase transition as second order. Unfortunately, this critical point is not alwaysaccessible. For example, for Ag-Cu, the theoretical critical point of the miscibility gap in the solidphase lies above the melting point of the material in the liquid phase (see Figure 1). I includesuch a material in our description. All these questions arise from well-established theories and aresubject to long-standing debates. The main purpose of this review is to shed some light on thesequestions from my personal perspective and to present a comprehensive model for first-orderphase transformations and microstructure evolution in homogeneous and heterogeneous media.Before starting with some basic relations and definitions, I give a brief overview about the actualdevelopments in this rapidly growing field of materials research.

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Cu concentration Cu concentrationV

olu

me

fre

e e

ne

rgy

(J

cm–

3)

Tem

pe

ratu

re (

°C)

0.2 0.4 0.6 0.8 1.00 0.2 0.4 0.6 0.8 1.00

a b

Liquid

Miscibility gap

Spinodaldecomposition

(unstable region)

0.1060.1060.106 0.3090.3090.3090.8400.8400.840

0.9610.9610.961 0.1060.1060.106 0.3090.3090.3090.8400.8400.840

0.9610.9610.961

Nucleation(metastable

region)

–6,0000

200

400

600

800

1,000

1,200

–5,600

–5,200

–4,800

–4,400

–4,000

Fcc-(Ag)Fcc-(Ag)

Fcc

Fcc-(Cu)Fcc-(Cu)

776

750

Figure 1(a) Calculated Ag-Cu phase diagram (solid red lines) with the miscibility gap (dashed line) and spinodal line(dotted-dashed line) of the fcc phase and (b) volume free energy of the fcc phase at 750◦C in the Ag-Au systemwith division of the spinodal region according to the thermodynamic parameters from Hayes et al. (2). Themolar volume is assumed to be 10 cm3 mol−1.

ACTUAL DEVELOPMENTS IN PHASE FIELD

Phase-field modeling has reached a state of maturity after more than 20 years of development.A number of review articles discussed different trends. In this journal in 2002, Chen (3) gavean excellent overview. He clearly explained the distinction between the two different notions ofphase fields: on the one hand, the idea of interface tracking and mimicking a sharp interface modelof motion and, on the other hand, the idea that the phase field corresponds to a physical orderparameter such as the composition field. That review focused on the school of Khachaturyan,which falls short in the present review. Moelans et al. (4) provided a comprehensive and tutorialintroduction along with a rich collection of the existing literature. Approaches to combine the twodifferent branches of phase field are exciting developments. In this regard I refer to the develop-ments of Kim et al. (5), Plapp and coworkers (6, 7), and my own contributions (8, 9). Basically,this development boils down to the problems of finding an appropriate interpolation scheme forthe order parameter in the interface and removing spurious effects by using an artificially widediffuse region for the interface in a mesoscopic simulation. This spurious interface energy andone strategy to avoid it are explained in detail below. Another important actual trend is the com-bination of phase-field models with the CALPHAD method (1), which has helped us realize theold dream of a consistent treatment of the kinetics of phase transformations, morphology, andalloy thermodynamics (10–15). This combination has become almost a standard today. Amongthe other exciting new developments is the development of phase-field models on the atomisticand microscopic scales. Elder and coworkers reported phase-field crystals in 2002 (for a review,see Reference 16). This new development has found many different applications, including theaplitude equation description by Spatschek & Karma (17) and the diffusive molecular dynamicsmodel by Li and coworkers (18). Another variant is the microscopic phase-field model, which restson the theory of microelasticity of Khachaturyan (19). It was first applied to dislocation movement(20–22); for a recent review, see Reference 23. Researchers have explored new frontiers of appli-cation such as glass formation (24), electrochemical reactions (25–30), and irradiation damage

www.annualreviews.org • Phase-Field Model at the Mesoscopic Scale 5.3


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(31). There are an overwhelming plurality of approaches and applications. As a last trend, let memention approaches that combine phase-field modeling of phase transformation and interfacemotion with involved models of bulk transport, as was developed in other communities, e.g., themechanics community [see the work of the French group around Forest, Busso, and Finel com-bining phase-field and crystal plasticity (32, 33)]. Additionally, diffusion coefficients are calculatedfrom chemical mobilities and a thermodynamic factor, and thus cross-effects in multicomponentalloys can be considered (34). To conclude this section, there are many future challenges, inparticular in solid-state transformations (35). Phase-field modeling, however, is an establishedtheoretical tool in materials science and engineering that closes the gap between the propertiesof single-phase materials and the properties of multiphase materials under consideration of mi-crostructure evolution during processing. Its potential has been demonstrated with many powerfulexamples, as is discussed above. Its future lies in truly quantitative application in combination withanalytical theory and/or well-controlled experimentation.

BASIC RELATIONS AND DEFINITIONS

Second- and First-Order Phase Transformations

A second-order phase transformation, as defined in Landau & Lifshitz (36), is a continuous tran-sition of the thermodynamic state of a material that allows for a clear distinction of phases. TheEnglish translation of Landau & Lifshitz’s book (p. 447) says: “Thus a phase transition of sec-ond kind is continuous in the sense that the state of the body changes continuously. It shouldbe emphasized, however, that the symmetry, of course, changes discontinuously at the transitionpoint. . . .” One speaks of broken symmetry below the transition point that makes the difference inthe states detectable. Very close to the transition point, also termed the critical point, the materialfluctuates between both states, and the length scale measuring the size of the microstructure ofthe individual phases is undefined. These characteristics are commonly referred to as criticality.The finding of scale invariance opened the wide field of critical phenomena [see the book series byDomb and colleagues (37, 38)], and the field was honored by Nobel Prizes in physics in 1982 (Ken-neth G. Wilson) and in 2003 (Alexei A. Abrikosov, Vitaly L. Ginzburg, and Anthony J. Leggett)(Landau himself passed away before his theory found the recognition it deserves). In contrast tothe overwhelming success of the theory of critical phenomena and renormalization group theories,numerical simulations of critical phenomena are difficult. This difficulty is, of course, due to thelack of a scale that would set a natural limit for the numerical discretization and the size of thedomain that is sufficient to resolve the long- and short-range fluctuations. Any numerical analysisof critical phenomena suffers from finite size effects (39). Nevertheless, the numerical model forsimulating a second-order, or critical, phase transition in a finite system is easily set up. We startout from a Ginzburg-Landau functional FGL as an integral over the domain �:

FGL =∫

�

r(T )φ2GL + κ(∇φGL)2 + uφ4

GL, 1.

with constants κ measuring the dissipation in the system and u setting the strength of regularizationof the order parameter φGL. r(T ) = r0

T −TcTc

measures the deviation of the temperature T fromthe critical point Tc with a positive constant r0. For T > Tc and r > 0, the minimum state ofthe functional is a stable uniform state with the order parameter < φGL >mf = 0, where <>mf

denotes a mean-field average neglecting fluctuations on a scale where κ ≈ 0. For T < Tc , wefind the well-known degenerate equilibrium states < φGL >mf = ±√ r

u . The range of applicabilityof the model in Equation 1, however, is extremely limited (40). In deviating from the criticalpoint, scale effects quickly become important. In analogy to the Ginzburg-Landau form (see

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Equation 1), we may investigate the Cahn-Hilliard form for spinodal decomposition (41). Outsidethe region of criticality, these models fail to give quantitative descriptions. There the justificationfor truncating the expansion in the order parameter and its gradients in Equation 1 is no longervalid. Furthermore, the parameter r is no longer a function of the deviation from the critical point.It reaches saturation, which may depend on many details of the system, and thus universality islost. A well-known example is critical opalescence in condensation of a gas passing through thecritical point. In deviating from the critical point, bulk phases are established, and the scale ofthe microstructure is large compared with the correlation length of the (noncritical) fluctuations.Variations of the order parameter condense into the interfaces between individual microstructureswith a thickness a in the range of several atomic distances. On this scale, the definition of an orderparameter is difficult: how to define, for example, (a) the mass density in the interface between aliquid and a gas or (b) the concentration of Ni atoms in the interface between the disordered γ

phase and the ordered γ ′ phase of a Ni-base superalloy? Without its physical interpretation, theorder parameter degenerates into an indicator function that distinguishes between bulk phases.The interface between these bulk phases has characteristics that cannot be derived from the valueof the indicator function. J.S. Langer first introduced this indicator function as a phase field in hisunpublished research notes from August, 1978. He noted,

The idea is to set up the simplest possible equation of motion of a phase-field φ(�x, t) which willidentify the phase [· · ·] at each point �x and time t. We don’t care whether these equations are a realisticmicroscopic model of any physical system—only that they reduce to our previous free boundary problemin appropriate limits.

The first mention of phase field in the printed literature was in Fix (42) in 1983. In 1986, Caginalp& Five (43) derived the sharp interface limit, demonstrating that phase-field models becomeindependent of the actual width of the interface η if η is small compared with the radius ofcurvature, r , of the interface between the two phases, i.e., if η � r . I follow this interpretation ofphase field below.

Mesoscopic Systems

A mesoscopic system refers to a body that consists of two (or more) parts distinguished by at leastone measurable quantity that shows a discontinuity on the mesoscopic scale. The mesoscopic scaleis defined as large compared with atomic distances in condensed matter but as small comparedwith the size of the bodies. The first criterion is needed to ensure a sufficient number of atomsin a volume of mesoscopic scale to ensure reliable statistics to define mesoscopic properties suchas concentration, temperature, pressure, and orientation. The region where this discontinuityoccurs is termed the grain boundary (between grains of the same solid phase), the phase boundary(between different condensed phases), or the surface (between a condensed phase and a gas);in general, this region is termed an interface. The simplest mesoscopic system, in the contextof this review, is a multicrystal that consists of one single solid phase, but the individual grainscan be distinguished by their orientation. As discussed above, we must introduce a minimummisorientation below which two grains must be considered as a single crystal (a homogeneoussystem). Below this minimum misorientation, we can define a so-called low-angle grain boundary,which is formed by a discrete arrangement of dislocations. This low-angle grain boundary is notconsidered here. Another simple example of a mesoscopic system is a mixture of a monoatomicliquid and its gas phase. Clearly this system must be considered as a homogeneous system insidethe spinodal region where the phase is unstable and inside the critical box around the critical pointof this system. Again, I introduce a minimum distance from the critical point beyond which the



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30 nm100 h/700°C

NiAl

Co 7 at%

CB

M23C6

γ'γ'

γ'

γ

Figure 2Elemental distribution map of a Ni-based superalloy (617B) sample annealed at 700◦C for 100 h. Adaptedfrom Reference 44 with permission.

system is considered as similar to a heterogeneous system, i.e., with clearly distinct propertiesbetween the two variants, here density, viscosity, and compressibility. This distance from thecritical point may be based on the correlation length of the fluctuation in the system, which isbelow our mesoscopic scale. The same assumption is adopted for systems with a miscibility gap orfor materials showing an ordering transition. In all these cases, the individual variants belong to thesame symmetry class, but individual variants can be clearly distinguished and can form an interfacewhen brought in contact. The actual value of the minimum mesoscopic scale may be under debatebut is here taken to be of the order of nanometers. Owing to recent advances in transmissionelectron microscopy (TEM) and atom probe tomography, atomic concentration measurementshave been reported down to the scale of several Angstroms. Figure 2 shows the atom distributionin a Ni-based superalloy with an M23C6 precipitate sitting in a microstructure of disordered γ

and ordered γ ′ by atom probe measurement. The averaged concentration profiles through theM23C6-γ interface and M23C6-γ ′ interface are shown in Figure 3. Those observations wereperformed by taking an average over several atomic layers parallel to the plane of the interface.However, close to the interface, the composition deviates considerably from its equilibrium valuein the bulk phases.

This observation leads to the definition of a chemical interface energy density σ c, in analogy toCahn & Hilliard’s (45) treatment of homogeneous phases, with the concentration c(n) as a functionof the normal distance to the interface:

σc =∫ + a

2

− a2

dn[ f (c (n)) + φα(n) f (c α) − φβ f (c β )]. 2.

Here the integration is in normal direction through the interface with an atomistic width a ofroughly 1 nm, f (c) is the free-energy density as a function of the composition, cα and cβ are theequilibrium concentrations in bulk, and φα and φβ are the phase fractions calculated by the leverrule, with c = φαc α +φβ c β and φβ = 1−φα . This model can be easily generalized to heterogeneous

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Co

nce

ntr

ati

on

(a

t%)

–5 –4 –3 –2 –1 0 1 2 3 4 5

a

0

2

4

6

8

10

12

14

Distance (nm)

Co

nce

ntr

ati

on

(a

t%)

–5 –4 –3 –2 –1 0 1 2 3 4 5

b

0

2

4

6

8

10

12

14

16

Distance (nm)

M23C6 γ M23C6 γ'

Mo

Co

BAl

Mo

Co

B

Al

1,000 h/700°C 1,000 h/700°C

Figure 3Atom probe tomography data and concentration profiles across M23C6-γ interfaces corresponding to the microstructure in Figure 2.(a) Concentration profiles across M23C6-γ interfaces, annealed at 700◦C for 1,000 h. (b) Corresponding profiles across the M23C6-γ ′interface showing B enrichment as well as pileup of Co at the M23C6-γ ′ interface. The error bars in the profiles correspond to 2σ

deviation errors. Adapted from Reference 44 with permission.

phases, where an effective mean-field free energy f = φα fα + φβ fβ is defined from the free-energydensities fα and fβ (see Figure 4):

σc =∫ + a

2

− a2

dn[φα fα(c ) + φβ fβ (c ) + φα f (c α) − φβ f (c β )], 3.

which reduces to Equation 2 if fα(c ) = fβ (c ) = f (c ). We employ this function below when thephase-field functional of a nanoscopic system is set up, i.e., when the concentration is used asan order parameter on the scale of atoms. We also need this function to clarify the distinctionbetween bulk and interface energies in a mesoscopic phase-field model (see below).

TWO-PHASE-FIELD MODEL FOR MESOSCOPIC SYSTEMS

Let us start with a simple two-phase system consisting of an α phase and a β phase. The bulk phasesare characterized by the phase-field variables φα = 1 and φβ = 1, respectively. The interface ischaracterized by the values of the phase fields, which lie between 0 and 1, subject to a constrainton the sum: φα + φβ = 1. The free energy, F2, of this two-phase system is then the integral overthe domain �:

F2 =∫

�

σαβ

ηIαβ + [ fα(c α) − fβ (c β )]hαβ . 4.

Here, σαβ is the interface energy between the phases or grains α and β. It will, in general, dependon the misorientation of the phases; the inclination of the interface with respect to the connectedgrains of the phases; and the concentration, temperature, and pressure. η is the interface width,which is taken to be constant and equal for all interfaces in the following. It is taken as a numericalparameter without physical interpretation. This setting rests on the demand of independence ofthe results of the calculations on η in a rigorous thin interface limit (which is discussed furtherbelow). fα(c α) and fβ (c β ) are the bulk free energies of the individual bulk phases. They may includestress-strain energy as well as electric or magnetic energy contributions. Here, however, they are



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Δf(c)

Δf(n)dn

fβ fα

f

a/2

–a/2∫

c(n)c(n)a

cβecβecα

ecαe

c n

Figure 4Scheme of the chemical interfacial energy caused by a continuous concentration through an interfacebetween two subphases that are separated by a miscibility gap. The left profile depicts the energy differencef compared with a lever-rule mixture as a function of concentration. The right profile depicts thecorresponding energy difference as a function of space. The integral of f over the interface atomistic widtha defines the interfacial energy.

treated, for simplicity, as dependent only on the phase concentrations cα and cβ . Both parts of thetotal free energy are clearly distinguished by their scaling in space. The interface energy has toscale with the area of the interface, and the bulk free energy has to scale with the volume of thephases. On the mesoscopic scale, where the atomistic width of the interface (a ≈ 1 nm) cannot beresolved, this scaling must be conserved under variation of the interface thickness η. Alternatively,one may state that the interfacial energy σαβ of a mesoscopic phase-field model is independentof η. Then Equation 4 ensures the correct scaling of the interface and bulk energies. An upperlimit for η defines the so-called thin interface limit, introduced by Karma (46). This limit is themaximum width of the interface in a numerical simulation that still allows for quantitative resultsfree from artifacts of the discretization. (This limit denotes a thin interface with finite width. Itmust not be confused with the so-called sharp interface limit, which is a mathematical limit η → 0in which the phase-field model can be compared with a corresponding sharp interface theory.)

Iαβ is a dimensionless interface operator, and hαβ is a dimensionless coupling function of theenergies. This operator and function may be taken differently in different models if they fulfillspecial conditions: Iαβ ≡ 0 in the bulk and the integral in the normal direction �n to the interface∫ ∞

−∞ Iαβdn = η. hαβ = h(φα) = 1 − h(φβ ) is a monotonic interpolation function with end valuesh(0) = 0 and h(1) = 1. This operator and function bear no physical information of its own besidesindicating the position of the interface and interpolating between bulk phases, respectively. Ofcourse, the interface operator also has the important tasks of propagating and stabilizing thephase-field contour and evaluating the local curvature of the interface in a numerical simulation.For this, I refer to the appendix of my previous review (47). In numerical applications, I preferthe so-called double-obstacle formulation because for this potential the phase-field contour has afinite width and the triple junctions show a natural energy penalty, which is not the case, e.g., forthe double-well potential. For illustrative reasons, let us use here the top-hat potential, which isconstant within the interface and has a linear interpolation function

I THαβ = −1

2[η2∇φα∇φβ + �(φα) − �(φβ )], 5.

hTH (φ) = φ. 6.

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Here, � is the step function �(φ) = 0 for φ ≤ 0; �(φ) = 1 for φ > 0. For completeness, anobstacle to cut off against unphysical values of φ < 0 and φ > 1 has to be added, although it isomitted here for readability. For details, see appendix A3 in Reference 47. Now let us return to thetreatment of the chemical interface energy (Equation 2) and its general form (Equation 3). Thetotal interface energy σαβ is divided into a chemical part, σ c, and a structural part, σ s. Inserting theinterface operator and coupling function into Equation 4, we have, with the chemical interfacialenergy (Equation 3),

F2 =∫

�

σs + σc

ηI THαβ + φβ fβ (c β ) − φα fα(c α)

=∫

�

{σs

η+ a

η[φα fα(c ) + φβ fβ (c ) + φα f (c α) − φβ f (c β )]

}I THαβ

+ φβ fβ (c β ) − φα fα(c α).

7.

Within the interface 0 < φ < 1, where the top-hat operator I THαβ is unity, all terms that depend

on the phase concentrations cα and cβ cancel out if the interface width is taken at the atomisticscale η = a . In this case, the model in Equation 7 reduces to an order parameter model in thespirit of Cahn & Hilliard (45), Khachaturyan (48), Wheeler et al. (49), and similar, on the basis ofthe continuous mixture concentration c. In any case, when η is taken as a numerical entity largerthan the atomistic scale a, one has to separate the bulk and interface contributions carefully, asdiscussed in more detail below.

As a general feature common to all phase-field models, below I discuss briefly the importanceand the consequences of a 1-D traveling-wave solution, i.e., the solution for a planar interfacetraveling with constant speed. The existence of such a solution imposes severe restrictions on thecoupling function h(φ), as described in the appendix of Reference 47. More important, however, isthe widely neglected condition that the driving force for the phase transformation gαβ be constantfor a traveling-wave solution to exist. Any variation of gαβ in the normal direction through theinterface will create a spurious interface energy. It will deform the shape of the interface in thenormal direction and will lead to a compression or expansion of the interface that is dependent onthe sign of the variation. In this situation, it is helpful to realize that the interface operators insidethe curly brackets of Equation 4 are both divergent to first order in 1

η. Because the interplay of these

operators is responsible for establishing the shape of the interface, a reduction of the interfacewidth η helps to minimize numerical artifacts. There are different strategies in the phase-fieldliterature for coping with this problem, and I cannot comment on them here. Personally, I preferan averaging procedure, as described in Reference 50 and implemented in the phase-field softwareOpenPhase (www.openphase.de). Nevertheless, all fields that are related to bulk properties showa strong variation over the interface in general. Examples are the mixture composition c and thestrain field ε in the solid state, which must be carefully decomposed into phase-dependent fieldsin a mesoscopic approach.

DIFFUSION IN A TWO-PHASE-FIELD MODEL

In the previous section, I show how the two-phase-field model becomes a physical order parametermodel in the limit η → a . Now let us go in the opposite direction to derive a two-phase-fieldmodel for diffusion on the mesoscopic scale by coarse graining a microscopic diffusion equationfor a binary alloy

c = ∇M ∇ ∂ f∂c

= ∇M ∇μ. 8.



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Here, M is the chemical mobility, f is the free-energy density, e.g., taken from Equation 7 in thelimit η = a , and μ is the diffusion potential μ = ∂ f

∂c . For constant M and a domain in 1-D normalto the interface, one has the Fourier transform

μ(�x) =∫

μ(�k)e−2π i �k�xd �k, μ(�k) =∫

μ(�x)e2π i �k�xd �x,

c (�x) = −4π2 M∫

k2μ(�k)e−2π i �k�xd �k.

9.

Now we split the Fourier modes into long- and short-range fluctuations with a cutoff wave vectork0 = 1

η. We are interested in keeping only the long-wavelength modes with |k| < k0, where η

sets the minimum resolution on the mesoscopic scale. This procedure leads to the coarse-grainedequation in each phase α and β in the �α domain and �β domain, respectively, where we havenonzero values of the phase-field variables φα �= 0 and φβ �= 0:

c α(�x) = −4π2 M∫

|�k|<k0

k2μ(�k)e−2π i �k�xd �k + P (μβ − μα); �x ∈ �α, 10.

c β (�x) = −4π2 M∫

|�k|<k0

k2μ(�k)e−2π i �k�xd �k + P (μα − μβ ); �x ∈ �β. 11.

The concentration is now split into the phase concentrations cα and cβ , and we have the chemicalpotentials μα = ∂ fα

∂cαand μβ = ∂ fβ

∂cβ. There is no more direct diffusion flux between the phases.

This flux is now replaced by the redistribution flux P (μβ − μα) in the interface where bothphases overlap. This flux is antisymmetric between the phases due to mass conservation. (Becausethe spatial information about the phases is lost by the integration over the short-wavelengthfluctuations, one has to recapture this information by the sign of the difference in the chemicalpotentials.) The average or the mixture concentrations are recovered by c = φαc α + φβ c β . Theredistribution flux is controlled by the interface permeability P, which can be formally defined asthe integral over the short-wavelength modes:

P = − 1(μα − μβ )

4π M∫

|�k|>k0

k2μ(�k)e−2π i �k�xd �k. 12.

P is thereby a function of the concentration and temperature in general and is proportional to thechemical mobility M, which is, in general, different in the interface than in bulk.

After coarse graining, only long-range concentration variations in bulk phases are explicitlyresolved. The steep concentration jump across the interface, which cannot be resolved on themesoscopic scale, is replaced by the concentration difference between the phase concentrations.Transforming back to real space, we have

c α = ∇M ∇μα + P (μβ − μα), 13.

c β = ∇M ∇μβ + P (μα − μβ ). 14.

In the case of a phase transformation, there will be, in addition, a correction taking into accountthe mass conservation during the transformation, which is explained in the next section. For now,let me conclude that the coarse-graining procedure, as outlined, has two important features: First,the mixture concentration c is split into the phase concentrations cα and cβ , which show gradientsonly on a scale that is large compared with the mesoscopic resolution η. Second, redistributionflux is introduced and takes into account the redistribution fluxes between the different phases asa local contribution that does not take into consideration the actual spatial arrangement of thephases on a microscopic scale. This flux is averaged over a volume that is large compared with the

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atomistic interface width a and that therefore has to be scaled by the ratio aη. This scaling leads to

a simple relation for the interface permeability as a first estimate based on bulk mobilities, whereμαβ denotes the potential jump across the interface in an off-equilibrium situation,

∇M ∇μ ∼ Ma2 μαβ → M

aη(μα − μβ ),

P = Maη

.15.

If a double-obstacle potential is used to stabilize the phase-field contours, as in the originalpublication (8), P can be replaced by the weighted average with the interface density 8φαφβ ,PDO = 8φαφβ

Maη

.

MULTIPHASE-FIELD MODEL

Let me briefly summarize the governing equations of the multiphase-field model for heteroge-neous systems based on the latest developments (8, 9). The total free-energy functional F as afunction of the phase fields {φα} and the phase concentrations {�c α} is defined as the integral overthe domain � of the interfacial energy density f intf and the chemical energy density f chem:

F =∫

�

{f int f + f c hem}

d�. 16.

The interfacial free-energy density f int f ({φα(x, t)}) at the location x within the domain � at timet, where N phases coexist, is defined by the gradients of the phase fields and the potential function(51),

f int f ({φα}) =N∑

α=1

N∑β=α+1

{4σαβ

η

[− η2

π2∇φα · ∇φβ + φαφβ

]}, 17.

and we have the natural constraintN∑

α=1

φα = 1. 18.

In general, σαβ may be a function of the phase concentration, the misorientation, or even the stress(52). Then the corresponding dependencies have to be considered in the functional derivatives.Here, we take the interface energy as a constant for simplicity. The chemical free-energy densityis

f c hem = f c hemαβ =

N∑γ=1

φγ fγ (�c γ ) +n−1∑i=1

λiαβ

[c iαβ − φαc i

α − φβ c iβ

], 19.

where the Lagrange parameter λiαβ has the meaning of a generalized chemical potential. It will be

evaluated such that the pair concentration in a junction �c αβ is conserved in a transformation fromphase α to phase β for arbitrary α and β (see model II in Reference 9). fα(�c α) is the bulk free-energy density of phase α evaluated for the phase composition �c α . This function can be directlyincorporated from a CALPHAD database or expanded in a suitable, thermodynamically consistentway. The composition vector runs from i = 1 to n − 1 for an n-component system.

The phase-field equation for an individual phase φα is defined as the sum of pairwise contri-butions (51, 53),

φα = −N∑

β=1

μαβ

N

{δ

δφα

− δ

δφβ

}Fαβ = 1

N

N∑β=1

ψαβ . 20.



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One calculates from the total free energy (Equation 16)

ψαβ = μαβ

⎧⎨⎩

⎡⎣σαβ (Iα − Iβ ) +

N∑γ=1,γ �=α,γ �=β

(σβγ − σαγ )Iγ

⎤⎦ + π2

4ηgαβ

⎫⎬⎭ , 21.

Iα ={∇2φα + π2

η2φα

}, 22.

where μαβ = 4η

π2 μαβ is the interfacial mobility between α and β, Iα is the capillary contributionfor phase α, and gαβ is the deviation of the bulk free energy from thermodynamic equilibriumor the driving force for the phase transformation,

gαβ ={

∂

∂φβ

− ∂

∂φα

}f chem. 23.

This driving force will be further detailed after the Lagrange function is evaluated from thedemand of mass conservation. After coarse graining, as discussed before, the diffusion equation issplit into a mesoscopic diffusion part in the bulk of each phase and the redistribution flux. Theredistribution part is considered locally within one reference volume,

c iα = −

N∑β=1

Piαβ

φα

∂

∂c iα

f chemαβ = −

N∑β=1

Piαβ [μi

α − λiαβ ], 24.

and the respective expression for the β phase. The Lagrange function can now be evaluated fromthe demand of mass conservation

0 = cαβ = φα cα + φβ c β + ψαβ (cα − c β ), 25.

λiαβ = φαμα + φβμβ

φα + φβ

− ψαβ (c iα − c i

β )

NPiαβ (φα + φβ )

. 26.

Adding the mesoscopic diffusion within the bulk phases driven by a gradient in the chemicalpotential μi

α = δ fαδc i

αand inserting Equation 26 into Equation 24 yield the diffusion equation with

the chemical mobilities M αi j in phase α,

φα c iα = ∇

⎛⎝φα

n−1∑j=1

M αi j ∇μ j

α

⎞⎠ +

N∑β=1

Piαβ

φαφβ

φα + φβ

(μiβ − μi

α) +N∑

β=1

φα

N (φα + φβ )ψαβ (c i

β − c iα).

27.The last term is necessary to ensure mass conservation in a moving interface. This term wasneglected in previous versions of the model, which assumed instantaneous redistribution in theinterface. This term is, however, important in cases in which the redistribution is sluggish and aphase transformation continues out of local equilibrium.

Now the driving force for the phase transformation (Equation 23) can be evaluated:

gαβ = fβ − fα −n−1∑i=1

λiαβ (c i

β − c iα) 28.

= fβ − fα −n−1∑i=1

[φαμ

iα + φβμi

β

φα + φβ

− ψαβ (c iα − c i

β )

NPiαβ (φα + φβ )

](c i

β − c iα). 29.

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Because the Lagrange function contains a contribution proportional to ψαβ , the interface mobilitywill be renormalized by a term that is dependent on the interface permeability or the chemicalmobility of the solute. After rearrangement, we end up with the final equations,

φα =N∑

β=1

Kαβ

N

⎧⎨⎩

⎡⎣σαβ (Iα − Iβ ) +

N∑γ=1,γ �=α,γ �=β

(σβγ − σαγ )Iγ

⎤⎦ + π2

4ηg phi

αβ

⎫⎬⎭, 30.

Kαβ = 4Nη(φα + φβ )μαβ

4Nη(φα + φβ ) + μαβπ2∑n−1

i=1(c i

α−c iβ )2

Piαβ

, 31.

g phiαβ = fβ − fα −

n−1∑i=1

φαμiα + φβμi

β

φα + φβ

(c iβ − c i

α). 32.

One very important aspect of the correction of the mobility (Equation 31) is that it allows one toinvestigate solute trapping and paraequilibrium transformations on a mesoscopic scale. Here, wemust distinguish between two limits. First, let us consider an interface between two phases withdifferent phase concentrations c i

α and c iβ . In this limit, diffusion is a must for a phase transformation

to proceed. If diffusion through the interface is very slow in at least one component i, Piαβ ≈ 0,

redistribution at the interface is prevented, and the transformation stops. This behavior is a result ofthe homogeneous average of the permeability parameters that enters the renormalized interfacemobility Kαβ in Equation 31. The second limit is complete solute trapping of a fast-movinginterface. When c i

α = c iβ , the influence of a finite permeability on Kαβ vanishes, and the phase

transformation can proceed in a massive manner. The transition between both limits is a delicatebalance between the different limits, which has to be investigated in more detail in the future.

EXAMPLE: SPINODAL DECOMPOSITION IN Ag-Cu

One large problem in the traditional formulation of the multiphase-field model with compositionssplit into phase concentrations on the basis of equilibrium lines (54) is the impossibility of consid-ering a miscibility gap. This problem comes from the ill-posed condition of a partition coefficientin the case of an opposite slope of the equilibrium lines of the different phases (or subphases).Starting from a Gibbs energy formulation, as in the approach outlined above (first proposed inReference 5), resolves this problem. Now the phase concentrations can be uniquely defined ir-respective of the type of the Gibbs energy function. This was demonstrated in Reference 8, butfor a diffusion couple example in which the initial composition of the couples was chosen outsidethe spinodal region. For completeness, a numerical procedure is outlined here. This procedureallows one to combine a Cahn-Hilliard (45)-type model of spinodal decomposition with the ac-tual mesoscopic phase-field model. The mesoscopic scale here is defined again by the minimumdiscretization of the numerical grid: x = 0.1 μm in the special example treated here. Belowthis scale, no useful information can be handled, and quantitative correct results can be expectedonly above a scale of ≈1 μm if we set the width of the phase-field contour to five grid cells. Thecalculations are performed on a 100 × 100–square grid in 2-D. We treat a system far from thecritical point; i.e., the physical interface width will be much below the resolution of the calculationon the nanometer scale, and we must be aware that the early stages of decomposition cannot befully resolved. The numerical procedure is split into three stages: (a) the initial decomposition thattreats the material as a single phase, (b) transition to the dual-phase state, and (c) final ripening. Thechemical mobility as a function of temperature (which will be set constant to 750◦C) and diffusionpotential are taken from the databases of References 2 and 55, respectively. For the phase diagram



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and Gibbs energy function, see Figure 1. The interfacial energy is set constant to σ = 1e−4 1c m2 .

In the present simulation, a constant phase-field mobility of μ = 50 c m4

Js is applied instead of therenormalized mobility corresponding to Equation 31. By using the chemical mobilities from Ref-erence 55 for the evaluation of Kαβ , it lies several orders of magnitude below the above-mentionedmobility. This value would considerably retard the dynamics of ripening against the dynamics ofdecomposition and diffusion. The following simulations neglect this retardation.

During the initial state of decomposition, there is only one diffusion equation of the mixtureconcentration of the binary alloy Ag-Cu with initial composition c (t = 0) = 0.5,

c = ∇M ∇(μ + κ∇2c ), 33.

where κ = σ xc s

is a measure of the interface energy. κ is chosen such that the interface energybecomes a constant σ as soon as two neighboring grid cells reach a composition difference equalto the composition difference c s between the spinodal concentrations during decomposition. Atthis point, the interfacial energy is handed over to the phase field in the second stage) (see below).We have the discretization of Equation 33 on a regular square grid with an explicit time-steppingscheme

c newi = c old

i + dt

⎧⎨⎩

N∑j

wi j M i j

x2i j

[(μ j − μi ) + κ

x2i j

(c oldj − c old

i )

]⎫⎬⎭ . 34.

Here, i is the index of the central grid point with N neighbors j. In 2-D, to which I restrictmyself in the following example, one chooses N = 8 with wi j = 2

3x2 and xi j = x fornext neighbors and wi j = 1

6x2 and xi j = √2x for diagonal neighbors. It is unnecessary

to resolve the fourth-order gradient in Equation 34 more closely as long as this calculation isrestricted to the initial state of decomposition, which falls below numerical resolution, but asuitable approximation of the interface energy created is needed. M i j is the chemical mobility ofthe atoms as an appropriate symmetric average between positions i and j. For simplicity, we treatit as composition independent, i.e., M i j = M . μi and μ j are evaluated at the individual pointsfor the local concentrations. The temperature and pressure are kept fixed in the simulations. Atthis stage, we need some considerations about the discretization. As discussed before, this casestudy is restricted to a discretization that is large compared with the physical interface width. Thedecomposition of the initially homogeneous concentration with some random perturbation willlead to a structure on the level of the numerical grid resolution. This is depicted in the left columnof Figure 5. The important point here is that this checkerboard microstructure is a steady-statesolution of the discretized Equation 34 if no interfaces are introduced. Let us briefly remain inthis situation. An alternative and often-used formulation of the diffusion equation is

c = ∇D∇c , 35.

with D = M d2 fd c 2 . The discretization in analogy to Equation 34 is

c newi = c old

i + dt

⎧⎨⎩

N∑j

wi j Di j (c j − c i )

⎫⎬⎭ . 36.

Dij obviously cannot be treated as a constant. Moreover, constructing a symmetric form, Di j =M2 [ d2 f

d c 2 |c i + d2 fd c 2 |c j ], does not solve the problem: The diffusion coefficient Dij becomes zero only if

both compositions are spinodal, i.e., at the inflection points d2 fd c 2 = 0. For our example of Ag-Cu

at 750◦C, these spinodal concentrations are c sα ≈ 0.309 and c sβ ≈ 0.84. A good solution can be

5.14 Steinbach


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Phase field

Cu concentration

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0P

ha

se fi

eld

or co

nce

ntra

tion

(a. u

.)

1.0e–9 s 1.0e–8 s 1.0e–7 s 2.5e–7 s 1.0e–6 s

1.0e–9 s 1.0e–8 s 1.0e–7 s 2.5e–7 s 1.0e–6 s

5 μm

Figure 5Decomposition of a homogeneous Ag-Cu alloy with 50% Cu as the initial condition. (Top row) Phase field. (Bottom row) Cuconcentration.

found only by a kind of finite-element integration between the grid points xi and xj,

Di j = 1x

∫ x j

xi

d 2 f (c (x))dc 2

d x. 37.

Using a linear shape function between the discrete nodes c (x) = c i − xi −xxi −x j

(c j − c i ), one easilycalculates, by substitution,

Di j = Mc i − c j

∫ c j

c i

d 2 f (c )dc 2

dc = Mc i − c j

∫ μ j

μi

dμ = Mμ j − μi

c i − c j. 38.

Inserting this diffusion coefficient into Equation 36 obviously leads back to Equation 34. The onlypossible way to discretize the diffusion in Equation 33 or 35 is thereby given by Equation 34 ifthe resolution cannot be chosen so fine as to avoid having the concentrations of two neighboringpoints lie on opposite sides of the miscibility gap. As soon as this situation appears during adecomposition process, it is natural to introduce interfaces in the second state.

The transition of a single-phase structure to a dual-phase structure starts at an individual gridpoint i as soon as the composition ci either falls below c sα or rises above c sβ . Then the pointis considered to be in phase α, φα = 1, or phase β, φβ = 1, respectively. The correspondingphase concentrations are c α = c or c β = c , respectively. The concentration of the phase withzero fraction is arbitrary but is taken for numerical consistency in the field of the counterphase.The corresponding phase-field distribution is depicted in Figure 5 (left column, top). It is acheckerboard pattern similar to that of the concentration in the bottom row but with top-hatvalues of 0 and 1. The solution of the phase field according to Equation 30 automatically leads toa coarsening of the structure and establishes well-resolved interfaces on a scale prescribed by thenumerical values of the interface thickness η. Because this initial pattern has, however, an averageinterface width η0 ≈ 2, the total interface energy of the system will be largely overestimated.Also, large numerical grid anisotropy will affect the process of coarsening. Whereas the lattercannot be remedied without significantly changing the grid resolution, the first can be safelycured by correcting the interfacial operator to the reduced interface width, which will be graduallyincreased with coarsening of the structure until a suitable interface width of η = 5x is reached.



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Cu

co

nce

ntr

ati

on

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1.0

z (μm)

Spinodalconcentrations

Equilibriumconcentrations

Figure 6Scan through the concentration of two spherical inclusions for the last time step at the right edge of thecalculation domain.

Figure 5 shows the phase-field and concentration evolution in the transition stage (see the secondand third columns). Finally, coarsening with the standard phase-field methodology sets in.

Even in the final state of this simulation, the concentration in α and β phases considerablydeviates from the equilibrium concentrations. Figure 6 plots the concentration along the scanline on the right boundary of the calculation domain. The scan crosses two spherical inclusions ofα and β phases in the counterphase. Due to the high curvature of roughly 1

1μm , the concentrationsare driven beyond the equilibrium concentrations to almost c α ≈ 0 and c β ≈ 1. In return,the concentrations of matrix-like regions are driven into the metastable region. Although thesecalculations show dramatically increased dynamics due to the limitation of a similar time-steppingcriterion of concentration and phase-field part, they demonstrate the applicability of the schemefor a realistic Gibbs free-energy function as given by a CALPHAD approximation.

As discussed above, the present treatment does not replace an involved study of the spinodaldecomposition at the critical point of an alloy or a fluid by a Cahn-Hilliard (45) model. This dis-cussion is meant to consider the homogeneous nucleation of an alloy prepared inside a miscibilitygap or inside a region of sublattice ordering, on a mesoscopic scale, or to treat the nucleation ofa subphase, e.g., during cooling from a single-phase region. On the mesoscopic scale, the initialstate of decomposition and nucleation cannot be resolved sufficiently. To address this issue, amultiscale treatment has to be applied, starting on the atomistic scale. The treatment as outlinedabove paves the way to such modeling on the mesoscopic side.

CONCLUSION

In this article, the phase-field method as a method for simulating the microstructural evolutionin materials is reviewed, with a special focus on the scale used to resolve the physical phenomena.Therefore, first, the scale-invariant phenomena close to a critical point were discussed. Apart fromthe critical point of a material with phases that belong to the same symmetry class, or materials thatbelong to different symmetry classes and therefore do not possess a critical point, we distinguishseparate scales for the interfaces (a) and the microstructure (η), the latter termed the mesoscopic

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scale. A phase-field model on the mesoscopic scale uses the numerical interface width as a parameterfor numerical convenience. The interface width should be as large as possible to allow for optimalnumerical performance but as small as necessary to resolve the microstructure and to not influencethe simulation results. Following this philosophy, which dates back to J.S. Langer (unpublishedresearch notes), a mesoscopic model with finite interface dissipation as recently developed bythe author and coworkers is outlined. This model allows for the treatment of microstructuralstates far from thermodynamic equilibrium. The model is applied to Ag-Cu prepared within amiscibility gap at elevated temperatures. The ability of the model to treat demixing and ripeningon the mesoscopic scale is demonstrated. For this situation, the initial scale of the microstructure,which lies on the atomistic scale, cannot be resolved properly. Resolution and discretization playa crucial role at the crossing points of the different scales. Here, numerical errors are unavoidablefor a single-scale method, and only rigorous multiscale analysis can quantitatively cope with thisproblem.

DISCLOSURE STATEMENT

The author is not aware of any affiliations, memberships, funding, or financial holdings that mightbe perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

The author would like to acknowledge financial support from ThyssenKrupp AG, Bayer Ma-terial Science AG, Salzgitter Mannesmann Forschung GmbH, Robert Bosch GmbH, BentelerStahl/Rohr GmbH, Bayer Technology Services GmbH, and the state of North Rhine–Westphalia,as well as the European Union in the framework of the ERDF.

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Phase-Field Model for Microstructure Evolution at the Mesoscopic Scale

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