1. Introduction

In the last decade, the capability to predict the solidifica-tion microstructure is considerably advanced due to the im-provement of the computer performance and the simulationmodels. Different techniques are used for the numericalsimulation of microstructure depending on the system size.For example, molecular dynamics and Monte Carlo meth-ods are favorable for nano-micro scale simulation, phase-field models are for micro-meso scale, and cellular automa-ta techniques are for meso-macro scale. Since the phase-field model satisfies the local equilibrium condition at lowinterface velocity and reproduced the solute trapping phe-nomena at high velocity regardless of the complexity of theinterface shape, it is expected as a powerful tool to simulatethe complex pattern evolution of the interface quantitative-ly. The phase-field model for solidification has been for-mulated by Langer1) based on the model C proposed byHalperin et al.2) and analyzed mathematically and thermo-dynamically.3,4)

Kobayashi5,6) showed the first example of the numericalsimulation using a phase-field model in which effects ofnoise and anisotropy on the dendrite shape were qualitative-ly analyzed. The calculation result was so impressive thatmany researchers rushed into the phase-field simulation.The phase-field model for binary alloys was soon pro-posed7) and was extended to the multi-phase8) and multi-component alloys9) step by step, also the convection effectwas introduced to the phase-field model in both pure mater-ial10–14) and binary alloy15–17) cases.

At the same time, the pure material model has been theo-retically improved to the thin interface limit model.18) In theconventional sharp interface limit, the calculation condition

is so unrealistic that no application examples are quantita-tively comparable to the analytical model or experimentalresults. Since the computational efficiency is improved re-markably and an arbitrary kinetic coefficient is available inthe thin interface model, three-dimensional calculations andthe comparison between the phase-field calculation and thatof solvability theory has become possible. The thin inter-face limit approach has been also applied to the binary al-loys19) and the phase-field model is established as a predic-tion tool for the solidification microstructure.20)

So far there is a review paper on the recent theoreticaldevelopment of the phase-field models21) but not on theirapplication examples. In this paper, we briefly review thephase-field models for solidification in section 2 and theirapplication examples such as free dendrite growth, direc-tional solidification, Ostwald ripening, interface-particle in-teraction and multi-phase-field simulations in section 3.Finally we mention future works of the phase-field model.

2. Governing Equations

2.1. Phase-field Model for Pure Materials

In the phase-field model, the state of the phase is repre-sented continuously by an order parameter, the phase-field,f . For example f51, f50 and a finite region in which0,f,1 represent solid, liquid and the interface, respective-ly. The time change of the phase-field is assumed to be pro-portional to the variation of the free energy functional.

................................(1)

where M is the phase-field mobility which is related to the

∂φ∂

δδφt

MF

5

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© 2001 ISIJ 1076

Review

Recent Advances in the Phase-field Model for Solidification

Machiko ODE, Seong Gyoon KIM1) and Toshio SUZUKI2)

Graduate student, The University of Tokyo, Tokyo 113-8656, Japan. 1) RASOM and Department of Materials Scienceand Engineering, Kunsan National University, Kunsan 573-701, Korea. 2) Department of Metallurgy, The University ofTokyo, Tokyo 113-8656, Japan.

(Received on March 16, 2001; accepted in final form on June 12, 2001)

The recent development of the phase-field models for solidification and their application examples arebriefly reviewed. The phase-field model is firstly proposed for pure material systems and then extended tobinary alloy, multi-phase and multi-component systems theoretically. Though the calculation conditions arelimited due to the sharp interface limit parameters in the early stage, it is widened in the thin interface limitmodel. The development of the phase-field model is summarized from a viewpoint of the formulation ofphase-field equation and parameters. The important studies and the latest results such as application exam-ples of free dendrite growth, directional solidification, Ostwald ripening, interface-particle interaction andmulti-phase simulation are mentioned. Finally future works of the phase-field model are prospected.

KEY WORDS: phase-field model; free dendrite growth; directional solidification; Ostwald ripening; particle–interface interaction; peritectic reaction; eutectic growth.

driving force for the interface. The Helmholtz free energyfunctional F has the form:

.................(2)

where e is the gradient term coefficient which is related tothe interface energy. It is recognized that the free energyfunctional has two different contributions. The first term inthe integral has a positive value only in the interface regionand related to the interface energy. The second term repre-sents the free energy density which is the sum of the freeenergies of solid, liquid and a double-well potential in theinterface region. The free energy density is expressed as:

f (f ,T)5h(f) f S1(12h(f)) f L1Wg (f) ...........(3)

h(f)5f3(10215f16f2) or h(f)5f2(322f) ....(4)

g(f)5f2(12f)2 ..............................(5)

where W is the height of the double-well potential, f S andf L are the free energies of the solid and liquid phase, re-spectively. Then the detailed equation of the phase-field isgiven as:

........(6)

In case of pure materials, a thermal diffusion equation issolved simultaneously. The latent heat generation term isadded to the conventional one,

....................(7)

where D is the thermal diffusivity, L is the latant heat and cp

is the specific heat.In the early phase-field model, the three parameters e , W

and M are selected arbitrarily.3–5) By assuming the equilibri-um condition within the interface T5Tm the phase-field pa-rameters are connected to physical properties.22) The e andW are related to the interface energy and interface width, Mis related to the interface kinetic coefficient, m and they areexpressed as Ref. 24).

.......(8)

where s is the interface energy, Tm is the melting point andd is the interface width, respectively. Since the parameter Mis derived assuming the temperature is constant within theinterface, small calculation mesh size or large undercoolingis required to neglect the temperature change though the in-terface region in the numerical simulation.23) Hence, the in-terface width should be negligible small comparing to cap-illary length theoretically at the sharp interface limit. Forthe computational efficiency, however, the value of the in-terface width is sometimes selected as large as possible be-yond the restriction. For example, the order of interfacewidth is 1028 m in pure material23) and alloy,24) respectively.

Karma and Rappel derived the phase-field mobility in the

thin interface limit to relax the restriction of the interfacewidth18) and expressed as follows:

......................(9)

The interface width, 1.631025 m is possible at thin inter-face limit of alloy20) to get experimentally comparable re-sults. Their thin interface limit model not only improvescomputer efficiency but also removes the limitation for thekinetic coefficient. They showed that the numerical calcula-tion of the dendrite shape using the phase-field modelagreed well with the solvability theory in both 2-D and 3-D.25–27)

Though Karma and Rappel assumed that the thermalconductivities in solid and liquid were equal, the thin inter-face limit model with the unequal conductivity has beenalso proposed by Almgren.28)

On the other hand, there is another approach to derive thephase-field equation. Since the temperature is assumed tobe constant in the free-energy functional based phase-fieldmodel, it has been pointed out that there is a thermodynam-ic contradiction in the model and the phase-field equation isderived assuming the positive entropy production.4) In themodel, the total entropy of the system has been expressedas:

..............(10)

where s(f , e) is the entropy density depending on the inter-nal energy density e and f . The phase-field parameters arederived similar way in the free energy functional model.22)

2.2. Phase-field Model for Binary Alloys

In the first phase-field model for alloys, the interface hasbeen assumed to be a region where the phase state changefrom one phase to another gradually. Hence the free energyin solid and liquid in Eq. (3) are the forms of composition-weighted mixture of the free energy of solute and solvent asfollows (WBM model).7)

f S5cf BS(T)1(12c) f A

S(T),

f L5cf BL(T)1(12c) f A

L(T) ....................(11)

Due to the sharp interface limit, the first WBM model didnot exhibit the solute trapping. Then the solute gradientterm was introduced into the free energy functional to re-produce the solute trapping phenomena.29) However, it canbe taken into account under the thin interface limit condi-tion as a relaxation of the solute flux though the interfacewithout the solute gradient term.19)

There is another definition for the interface composition.Steinbach et al. considered the interface as a fraction-weighted mixture of the solid and liquid with differentcompositions and free energy. The compositions within theinterface have been determined to keep the equilibrium par-tition coefficient cS5kecL (0,f,1).30) Kim et al. improved

S s e dVV

5 2 11

2 02 2

ε φ φ∇

∫ ( , )

Kh h

d∫1

0 1

15

2

2

φ φφ φ

φ( )( ( ))

( )

ML

T

L

D cK

m k P

−

−

⋅

12 1

1

45 1

εσ µ

δ,

WT T

MT

L5 5 5

3

2

6 2

6 2

22σ

δε

σδ µ

δm m

m, ,

∂∂

φ∂φ∂

T

tD T h

L

c t5 1∇2 9( )

P

∂φ∂

ε φ φ φt

M h f f WgL S5 1 2 2[ ( ){ } ( )]2 2∇ 9 9

F f T dVV

5 11

22 2

ε φ φ∇

∫ ( , )

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this idea and the compositions of solid and liquid at the in-terface are determined to have the same chemical potential(KKS model).19)

f S5cS f BS(T)1(12cS) f A

S(T),

f L5cL f BL(T)1(12cL) f A

L(T),

c5h(f)cS1(12h(f))cL, mS(cS(x, t))5mL(cL(x, t)) ......(12)

Since the free energy density in KKS model correspondsto the common tangent rule, no excess chemical free energywhich prevents adopting the large interface width appears.They have also derived the phase-field mobility in the thininterface limit for alloys.

In case of alloys, a solute diffusion equation is solved si-multaneously.

......................(13)

where Dc is the solute diffusivety and fc and fcc are the firstand the second derivatives of the free energy density. Thesolute diffusion equation assures that the time change of thesolute is proportional to the gradient of the chemical poten-tial fc, fcc is added to guarantee a constant diffusivities inboth the bulk solid and liquid phases.

Phase-field parameters, e and W, are the same as the onesin the pure material case in the sharp interface limit. Thephase-field mobility in the thin interface limit is also de-rived by Kim et al.19) in the same manner as the pure mater-ial case.

,

...(14)

Since the mobility is the first-order correction in that of thesharp interface limit, the accuracy increase with the largervalue of the partition coefficient. The above mobility is de-rived in the 1-D situation but its definition in 2-D and 3-Dis still under active discussion.28, 31)

Strictly speaking, the phase-field equation for alloysshould also be derived to assure the positive entropy pro-duction with the entropy functional of the form:32)

............................................(15)

The thermodynamically consistent model is ideal to adoptthe open or non-isothermal system theoretically. The freeenergy functional based phase-field equation is, however,also applicable in case of metric alloys because the thermaldiffusivity is much larger than the solute one in metal sys-tems and the temperature can be regard as constant locally.

2.3. Multi-phase-field Model and Multi-componentModel

One of the advantages of the phase-field model is that itis extended to the multi-phase and multi-component sys-tems straightforwardly. In the multi-phase-field model, eachphase is distinguished by each order parameter f1, f2,…, fn

and the order parameter can be considered as a volumefraction of the phase of the reference number.

.................................(16)

In the n-phase-field system, the free energy functional iswritten as:8)

............(17)

The phase-field parameters are determined by the same wayas in a single phase model.

In the multi-component systems, the free energy of solidand liquid are written as:

.................................(18)

...............(19)

The cross terms in the solute diffusion equation are stillunder discussion and only a dilute multi-component alloymodel is proposed9) and the phase-field mobility in the thininterface limit is also available with the infinite kinetic co-efficient.

............(20)

There is another attempt to simulate multi-componentsystems with the phase-field model.33) Grafe et al. introducethe thermodynamics database such as Thermo-Calc into thephase-field equation and Dictra software is adapted to de-termine the diffusion matrix in the solute field equation.

3. Application Examples

3.1. Free Dendrite Growth

Dendritic crystal growth analysis is popular in the phase-field studies and there are lots of application examples forpure materials,18,23,25–27,34–39) pure materials with convec-tion,10–13) binary alloys24,40,41) and ternary alloys.9)

The first numerical simulation using a phase-field modelwas also the study of the free dendrite growth in a pure material.5,6) Figure 1 shows the first numerical example of

MW D

c cji

j j j

j

n− ∑1

3

12

15

5

ε

σζ ( , )L

eSe

i iS L with in the interface)5 (µ µ

f c f T f c f Ti i

i

n

i i

i

nS S S L L L5 5

5 5

( ), ( ) 1 1

∑ ∑

ci

i

n

511=

∑

h f T W p dVi i ik i k1 1

φ φ φ φ( ) ( , ) ( , )

F ik i k k i

k

i

i

n

5 ∇ 2 ∇ 1

55

∑∑ 1

22 2

11

ε φ φ φ φV∫

φi

i

n

511=

∑

s e c dV1 φ

( , , )

S c ec eV

5 2 1 1 11

2

1

2

1

202 2

02 2

02 2

ε φ ε ε∇ ∇ ∇

∫

ς

φ φφ φ

φφ φ

( , ) ( ) ( )( )

( )[ ( )]

[ ( )] ( ) ( ) ( ) ( )

c c f c f c c c

h h

h f c h f c

d

Se

Le

ccS

Se

ccL

Le

Le

Se

ccS

Se

ccL

Le

5 2

32

2 1 2

2

0 0

0 00

10

0 0

1

1 1∫ ⋅

MRT

V

k

m D Wc c−

12 1 1

25

21

εσ µ

ες

m

e

eL

Se

Le( , )

∂∂

φc

t

D

ff5∇⋅ ∇

c

ccc

( )

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the free dendrite growth using a phase-field model byKobayashi.

The anisotropy introduced in gradient term coefficient asfollows.

where n is the amplitude of the anisotropy and k representsthe k-hold symmetry. Noise is introduced to reproducewell-developed side branches and added into the phase-fieldequation. By varying the value of the phase-field parame-ters arbitrary, he showed the effect of the anisotropy andnoise in the shape of the dendrite qualitatively. Wheeler etal. adopted the phase-field parameters which were deter-mined from physical properties and studied the dendrite tipshape more quantitatively. They also pointed out the calcu-lation result changed depending on the interface thicknessbecause of the sharp interface limit.23) Then Karma andRappel showed that phase-field simulation agreed with theprediction of the solvability theory in 2-D and 3-D byproposing the thin interface limit.25–27)

Thermal noise has been introduced for the quantitativeanalysis of side brunch.36) By solving Navier-Stokes equa-tion simultaneously, the convection effect has been alsostudied using the thin interface limit model.10,11,13) To re-duce the computational time, adaptive mesh techniqueshave been proposed.34,37)

Though there are numerical examples of the free dendritegrowth in alloys,24,40) almost all the results are qualitative.There is one example of the prediction of the secondaryarm spacing, which is comparable to the experimentaldata41) Figure 2 shows that the calculated arm spacingagrees well with that of the experiment.

3.2. Directional Solidification

The cellular pattern formation in the directional solidifi-cation is also studied using a phase-field model. Losert etal. studied the range of the imposed perturbation wave-length from singlet to doublet pattern formation in a purematerial using the thin interface limit model.42) Contishowed so-called banded structure in rapidly solidified bi-nary alloys43) and Boettinger et al. showed that the segrega-tion pattern and the breakdown of planar interfaces into cellular structures near the absolute stability interface ve-locity.44) They showed the dynamics of the transition of theinterface pattern that were observed in the experiment qual-itatively. However, there is few quantitative informationabout the pattern formation. In other word, though the cal-culated microstructure looks like the experimental one, thesolidification conditions such as temperature gradient andinterface velocity are restricted. On the other hand, Kim etal. showed the calculated cellular pattern which agreedquantitatively with the experimental data in a binary sys-tem.20) Figure 3 shows that the calculated cellular pat-tern using the same condition as the one used in the experi-ment.45–47) As pointed out before, their model is not perfectthermodynamically but computationally efficient, and givesus quantitative predictions of the microstructure.

In the directional solidification of alloys, solute-trappingphenomena is also examined. Since the solute trapping phe-nomena did not emerged in the first alloy phase-field model

ε ε ν θ θ∂φ ∂∂φ ∂

5 1 5( cos ), tan/

/1 k

y

x

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Fig. 1. The first numerical phase-field simulation. The an-isotropy is introduced in the driving force for the inter-face. The amplitude parameter d is (a) 0.2 (b) 0.3 (c) 0.5respectively.63) The effect of the anisotropy and the com-petitive growth of the side branches are nicely repro-duced. (By courtesy of R. Kobayashi)

Fig. 2. The secondary arm spacing vs. local solidification time inAl–4.5wt%Cu alloy.41) The calculated arm spacing andtangent shows good agreement with that of experiment.64)

Fig. 3. The directional solidification of the CBr4–8 mol%C2Cl6

alloy.20) The {111} plane of the crystal is parallel to theplane of the thin film and the k110l direction is parallel tothe pulling direction; (a) V510 mm/s, (b) V515 mm/s and(c) V520 mm/s and G50.83104 K/m. The calculation re-sults show good agreement with the experimentaldata.45–47)

due to the sharp interface limit, the solute gradient termwas reconsidered29,48) However, since the chemical potentialgradient exists across the interface under the thin interfacelimit condition, the solute trapping is reproduced naturallywithout the term49,19) It is shown that the calculation resultsagree with the prediction by the Aziz’s equation.

3.3. Ostwald Ripening

Since the phase-field model can take the interaction be-tween the solute fields around each particle into account, itis very useful to analyze the Ostwald ripening problem.Figure 4 shows that the solute fields around the particlesduring the Ostwald ripening. The first Ostwald ripeninganalysis adopted very small calculation mesh size and thevolume fraction changed during the coarsening process bythe curvature effect.50) Diepers et al. studied the convectioneffect on the Ostwald ripening process qualitatively.Vaithyanathan and Chen studied the difference in the simu-lation result between 2-D and 3-D simulation using thephase-field model for solid solution.51)

3.4. Interface-particle Problem

It is difficult to simulate the interface shape around an insoluble particle with the conventional shape interfacemodel. Even if local equilibrium is assumed, the interfaceshape is still remained to be undetermined. The first analy-sis for the interface shape around the insoluble particle inalloy is studied by Ode et al. Then they showed the timehistory of the interface shape during the particle pushing/engulfent process and estimate the critical velocity for par-ticle pushing/engulfment.52,53) Figure 5 shows that time his-tory of the particle pushing/engulfment by the interface ofFe–C alloy.

3.5. Application Example Using Multi-phase-fieldModel

So far only the multi-phase-field model can simulate thecomplex microstructure of the eutectic,8,54–56) peritec-tic8,54,55,57,58) and monotectic alloys.17) The idea of the non-single phase-field model has been proposed by Wheeler etal. for eutectic growth56) and Steinbach et al. for multi-phase-field model8) at almost the same time. The variouseutectic lamella structure is reproduced54,55) and the peritec-tic phase growth behind the dendrite tip region is nicely reproduced.58) Figure 6 shows that the peritectic phasegrowth during the free dendrite growth and Fig. 7 showsthat the rod structure growth. In the multi- phase fieldmodel the quantitative comparsion with the experimantaldata also becomes possible. Figure 8 shows eutectic lamel-la structure changes depending on the composition.

Multi-phase-field models are also applied to the graingrowth analysis8,59) Each phase-field represents the eachgrain with a specific orientation. However, a single phase-field model with an orientation contribution term in the free energy functional, so-called vector-valued phase-field

ISIJ International, Vol. 41 (2001), No. 10

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Fig. 4. The interaction of the solute fields of each particle duringOstwald ripening process in Al–4.0wt%Cu alloy.65) Themelting and disappearance of small particles, and thegrowth and agglomeration of larger particles are qualita-tively reproduced. (a) 1.0 s, (b) 20 s, (c) 40 s, (d) 60 s.

Fig. 5. The pushing/engulfment of an alumina prticle by the in-terface in Fe–0.5 mol%C alloy. The forces acting on theparticle are estimated by the interface shape at each time-step.66) (a) pushing, (b) engulfment.

Fig. 6. Peritectic transformation during directional solidificationin Fe–0.36wt%C alloy (T522 K/s, =T52250 K/cm).The austenite nucleation is imposed at the dendrite sur-face when the temperature is below the peritectic one.The austenite growth by consuming ferrite behind thedendrite tip is nicely reproduced.67,68) (By courtesy of I.Steinbach)

model, is also proposed for grain boundary analysis. Thevector-valued phase-field model can reproduce grain orien-tation arbitrary and the grain coalescence.60–62)

4. Concluding Remarks

The recent developments of the phase-field models andtheir application examples are briefly reviewed. So far, noother method can be compared with phase-field models forsimulating the complex interface evolution in not only purematerials but also in multi-phase and multi-component sys-tems. Since there are application trials for solid transforma-tion, evaporation and electrodeposition processes, thephase-field models will be applied to almost all the forma-tion processes of microstructure in the near future.

The key point of the phase-field model for solidificationwill be the quantitative prediction of the microstructure forindustrial materials. The multi-phase and multi-componetmodels should be extended to simulate the microstructureevolution from the crystallization of primary and secondaryphases to the solid state transformation continuously andquantitatively. Since the basic idea has been already accom-plished, treatment of nucleation, coupling with a thermody-namic database and timesaving techniques should be inves-tigated in detail. There are a few models to produce a newphase seed such as coupling with the classic nucleation the-ory, however, it needs more examination to obtain reliableresults. The trial to couple the thermodynamic databasewith the phase-field calculation has started recently and it isnecessary to widen the scope of the application to industrialmulti-phase and multi-component alloys. Now, the most se-rious problem in a practical use of the phase-field model isthe computational time. Timesaving algorithm and parallelcomputing technique are expected for the further applica-tion.

Acknowledgements

The authors are grateful to R. Kobayashi of HokkaidoUniversity and I. Stainbach of ACCESS e.V. for their fig-ures.

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