Phase-field model for multicomponent alloy solidification

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0022-0248/$ - se

doi:10.1016/j.jcr

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Journal of Crystal Growth 274 (2005) 281–293

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Phase-field model for multicomponent alloy solidification

Pil-Ryung Chaa,�, Dong-Hee Yeonb, Jong-Kyu Yoonb

aSchool of Advanced Materials Engineering, Kookmin University, 861-1, Chongnung-dong, Songbuk-gu, Seoul 136-702, Republic of KoreabSchool of Materials Science and Engineering, Seoul National University, Seoul 151-742, Republic of Korea

Received 21 May 2004; accepted 1 October 2004

Communicated by G.B. McFadden

Available online 18 November 2004

Abstract

A phase-field model is presented for studying the solidification of multicomponent alloys, which can consider the

different diffusion mechanisms between the substitutional and the interstitial solutes appropriately. By defining the

interfacial region as a mixture of solid and liquid with the same inter-diffusion potential and determining and matching

the phase-field parameters to the alloy properties at a thin-interface limit condition, the limit of interface thickness due

to the contribution of the chemical free energy density in the interfacial region is remarkably relieved and a large-scale

calculation domain is permitted. One-dimensional numerical calculations for the solidification of an dilute Fe–Mn–C

alloy system demonstrate that the model could provide a good description for the equilibrium and kinetic properties for

isothermal solidification of multicomponent alloys. Two-dimensional computations for the evolution of large-scale

dendritic microstructures is performed and solute distributions in both the solid and liquid phase and the dendritic

shapes are compared at different temperatures.

r 2004 Elsevier B.V. All rights reserved.

PACS: 64.70.Dv; 81.30.Fb; 81.10.Aj; 05.70.Ln

Keywords: A1. Growth models; A1. Morphological instability; A1. Phase-field model; A1. Solidification; B1. Multicomponent alloys

1. Introduction

The phase-field model has attracted manyscientists due to its ability in describing thecomplex pattern formations in phase transitions

e front matter r 2004 Elsevier B.V. All rights reserve

ysgro.2004.10.002

ng author. Tel.: +8229104656;

284.

ss: [email protected] (P.-R. Cha).

such as dendrites. It is a useful method forrealistically simulating microstructural evolutioninvolving diffusion, coarsening of dendrites andthe curvature and kinetic effects on the movingsolid–liquid interface. It is efficient, especially innumerical treatment, because all the governingequations are written in a unified form withoutdistinguishing the interface from the solid or fromthe liquid phase. In the model, the phase-field,

d.

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P.-R. Cha et al. / Journal of Crystal Growth 274 (2005) 281–293282

fð~r; tÞ; characterizes the physical state of thesystem at each position and time: f ¼ 1 for thesolid, f ¼ 0 for the liquid, and 0ofo1 at theinterface. The phase-field variable, f; changessteeply but smoothly at the solid–liquid interfaceregion, which avoids direct tracking of the inter-face position. Therefore, the model can beregarded as a type of a diffuse interface model,which assumes that the interface has a finitethickness and that the physical properties of thesystem vary smoothly through the interface.A diffuse interface model was first developed by

Van der Waals [1], who considered fluid density asan order parameter. Thereafter, by the mid of1980s, the diffuse interface model was applied tothe equilibrium properties of the interface [3],antiphase boundary migration by curvature [4],and to the second-order transition [2] but not tothe first-order transition. Langer [5] proposed thatthe diffuse interface model could be applied tosolidification phenomena. By using the singularperturbation method [6], Caginalp [7] proved thatthe phase-field model could be reduced to theStefan problem in the limit that the thickness ofthe interface approaches zero and Kobayashi [8]studied the dendritic growth of a pure melt byusing the phase-field model for pure materials. Themodels which were proposed for simulatingdendritic growth in pure undercooled melt [8–13]have been extended to modeling of binary alloysolidification [14–23] and recently multicomponent

alloy solidification [24].The phase-field models for alloy solidification

may be classified into four groups depending onthe definition of free energy density for interfacialregion and how they were derived [23]. The first isa model by Wheeler, Boettinger, and McFadden(hereafter abbreviated by WBM) model[14–16,20], which has been used most widely. InWBM model, the interfacial region is assumed tobe a mixture of solid and liquid with the samecomposition, but with different chemical poten-tials, which leads a certain limit in the interfacethickness due to chemical energy contribution tothe interfacial energy and this limitation makes thelarge-scale simulation difficult [20]. Caginalp andXie [18] have also proposed a similar model. Thesecond is a model by Steinbach and coworkers

[19], which uses a different definition that theinterfacial region is assumed to be a mixture ofsolid and liquid with different compositions, butconstant in their ratio. It has less limit in theinterface thickness but the model is thermodyna-mically correct for a dilute alloy only. The thirdmodel has been developed by Losert and cow-orkers [21], and Lowen and coworkers [22], whichis based on two unrealistic assumptions: One isthat the liquidus and solidus lines in the phasediagram are assumed to be parallel and the otherthat the solute diffusivity is constant in the wholespace of the system. The fourth was proposed byKim and coworkers [23] which is free from thelimit for the interface thickness in the WBM model[14–16] by assuming the interfacial region as amixture of solid and liquid with the same chemicalpotential, but with different compositions.In studying alloy solidification, however, there

still remains a major problem with the modelsdiscussed above [14–16,19–21,23]. The problem isthat all the existing models are restricted to binary

alloys with substitutional solutes. For an alloywith interstitial solutes, these models are notapplicable because a mole fraction as a concentra-tion variable can cause a serious error [24,25]. Itcan be said that the application of the phase-fieldmodel for solidification to practical technology isclosely linked to our ability to model microstruc-tural development in multicomponent alloys (threeor more components) because the most alloys forpractical use are multicomponent. The existingmodels [26,27], which are the sharp-interfacemodel and can describe the multicomponent alloysolidification, are applied to only one-dimensionalproblems (for example, DICTRA code) and hencethese models can not describe the pattern forma-tion, solute trapping, solute drag and para-equilibrium phenomena in multicomponent alloysolidification. In addition, there have been a fewresearches for the solidification of multicomponentalloys (three or more components) [28,29], butmost of them have the limit of the interfacethickness or can be applied only to substitutionalalloy systems. Therefore, it is very important todevelop a phase-field model for the solidificationof multicomponent alloy systems with both sub-stitutional and interstitial elements.

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P.-R. Cha et al. / Journal of Crystal Growth 274 (2005) 281–293 283

Recently, Cha and coworkers [24] presented aphase-field model for the isothermal solidificationof multicomponent alloys. However, the modelhas a severe limit of the interface thickness due tothe variation of the chemical free energy in theinterfacial region. For example, the interfacethickness should be less than 85 nm for an Fe—1.0 at% Mn—0.5 at% C alloy system [24]. Thisthickness limit constrains the size of calculationdomain and makes it very difficult numericalsimulation at a low undercooling, where thecurvature of the dendrite tip would become smalland thus a large computation domain would berequired.In this study, we present a phase-field model for

multicomponent alloy solidification which relievesthe limit of the interface thickness. For thispurpose, this work is organized as follows. InSection 2, a phase-field model is developed formulticomponent alloys with both substitutionaland interstitial alloy elements. The parameters inthe phase-field equation are then matched withalloy physical properties and the phase-fieldmobility is determined at the thin interface limit

where the inability to deal with local equilibriumand the restriction on the calculation domain arerelieved. In multicomponent alloy solidification,no general relationships exist among temperature,interface velocity and concentration [30]. There-fore, the phase-field mobility is determined usingthe chemical rate theory for solidification[30–33,35]. In Section 3, the evolution equationsof the phase-field and concentration fields arerewritten in the dilute solution limit and thesteady-state solutions of the concentration profilesacross the interface are deduced. In Section 4,numerical calculations for one-dimensional solidi-fication of a ternary alloy and for the large-scaletwo-dimensional dendritic growth of diluteFe–Mn–C alloy system are presented.

2. Phase-field model to multicomponent alloys

Most phase field approaches for alloy solidifica-tion using a mole fraction as the concentrationvariable [14–23] are not useful for an alloy systemwith interstitial solutes, due to the difference in

diffusion mechanism between an interstitial and asubstitutional species (see [24,25] for detaileddiscussion). By employing the number of molesin a unit volume as the concentration variable anddefining the interfacial region as a mixture of solidand liquid with the same chemical potential butwith different compositions, a new phase-fieldmodel is here presented for studying the solidifica-tion of a multicomponent alloy system containingboth interstitial and substitutional alloying ele-ments, which does not suffer from the limit ofinterface thickness due to the contribution ofchemical free energy in the interfacial region[23,24].Consider a solution with n components, which

may exist as either a solid or a liquid in a domain,O: We now choose the number of moles of the ithcomponent in a unit volume, ci; as the concentra-tion variable. For mathematical simplicity, how-ever, the following assumptions are made: (i) boththe solvent and all substitutional alloying elementshave an identical partial molar volume, D; (ii) thereis no contribution of interstitial element to thealloy volume, (iii) the difference in the molarvolume between the solid and liquid phase isneglected, and (iv) the amount of substitutionalvacancies are also negligible.As the number of moles of the substitutional

and interstitial lattice sites per unit volume isconserved, respectively, the new concentrationvariables have the following relationships:P

i2S ci ¼ 1=D andP

i2I ci þ cIE ¼ const:; wherethe symbols, S and I ; indicate a substitutional andan interstitial element, respectively, and cIE is themole number of empty interstitial sites per unitvolume. If the solvent is designated as the nthelement, the concentrations of both the solventand the empty interstitial sites are convenientlyremoved, leaving the number of the independentconcentration variables at n � 1:The total free energy of the system is defined as

Fðf; c1; . . . ; cn�1Þ

¼

ZOdO f ðf; c1; . . . ; cn�1Þ þ

�2

2jrfj2

� �; ð1Þ

where f is the free energy density of the system, �represents gradient corrections to the free energy

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density, and f is a phase field ranging from zero inthe liquid to one in the solid.In this work, the definition of the interfacial

region proposed by Kim et al. [23] is adopted toremove the limitation of the interface thicknessdue to the contribution of the chemical free energyin the interfacial region. Any point within theinterface region is considered as a mixture of solidand liquid with a same inter-diffusion potential.Then, the free energy density and concentration ofith component are expressed as

f ðf; c1; . . . ; cn�1Þ

¼ wgðfÞ þ hðfÞf SðcS1 ; . . . ; cSn�1Þ

þ ½1� hðfÞf LðcL1 ; . . . ; cLn�1Þ ð2Þ

and

ci ¼ hðfÞcSi þ ½1� hðfÞcLi ; (3)

with

f ScSiðcS1 ; . . . ; c

Sn�1Þ ¼ f LcL

iðcL1 ; . . . ; c

Ln�1Þ; (4)

where f S and f L are the free energy densities of thehomogeneous solid and liquid phases, respectively,and cSi and cLi are the mole numbers of the ithcomponent in the solid and liquid phases, respec-tively. f ScS

iand f LcL

iare qf S=qcSi and qf L=qcLi ;

respectively. gðfÞ is a double-well potential, forexample, one of the simplest forms is f2

ð1� fÞ2

and w is its height. The interpolation function hðfÞrepresents a normalized area under the potential,gðfÞ:In order to derive a kinetic model, it is assumed

that the system evolves in time so that its total freeenergy decreases monotonically. The evolutionequation for phase field becomes:

1

M

qfqt

¼ �dFdf

¼ r � �2rf� f f

¼ r � �2rf� wg0ðfÞ þ h0ðfÞ

f L � f S �Xn�1i¼1

ðcLi � cSi ÞfLcL

i

" #; ð5Þ

while the concentration of the kth componentshould evolve with:

qck

qt¼ r �

Xn�1i¼1

MkirdFdci

¼ r �Xn�1i¼1

Mkirf ci

¼ r �Xn�1i¼1

Xn�1j¼1

Mkif cicjrcj

þ r �Xn�1i¼1

Mkif cifrf; ð6Þ

where M is the phase-field mobility and Mki

represents a component of the mobility matrix ofsolute atoms, ~M: f f; f ci

; f cif; and f cicjare qf =qf;

qf =qci; q2f =qciqf; and q2f =qciqcj ; respectively.Mki is determined from the diffusivity matrix,Dkj ¼

Pn�1i¼1 Mkif cicj

: h0ðfÞ and g0ðfÞ represent

dhðfÞ=df and dgðfÞ=df; respectively.In order to find the composition (c0kðxÞ) and

phase field (f0ðxÞ) profiles at equilibrium and the

relationships between material properties andphase-field parameters (� and w) of the presentphase-field model, a one-dimensional solidificationproblem is considered at equilibrium (see [24] fordetailed discussion). Under boundary conditionsof f0

¼ 1 (solid) at x ! �1 and f0¼ 0 (liquid)

at x ! þ1; Eqs. (5) and (6) for a one-dimensional

equilibrium state yield using the same approachintroduced in Ref. [24]:

f0ðxÞ ¼

1

21� tanh

ffiffiffiffiw

pffiffiffi2

p�

x

� �(7)

and

c0i ¼ hðf0ðxÞÞcS;ei þ ½1� hðf0

ðxÞÞcL;ei ; (8)

where cL;ei and cS;ei are the ith concentrations in theliquid and solid phases at equilibrium, respec-tively. Using the composition and phase-fieldprofiles at equilibrium, direct evaluation of theinterface energy, s; and interface thickness, 2l;gives

s ¼�

ffiffiffiffiw

p

3ffiffiffi2

p ; (9)

2l ¼ affiffiffi2

p �ffiffiffiffiw

p ; (10)

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where a is a constant which is dependent on thedefinition of the interface thickness, e.g., a ’ 2:2when f0 changes from 0.1 to 0.9 at �loxol; anda ’ 2:94 when f0 changes from 0.05 to 0.95[24,25]. It should be noted that the parameterrelationships (9) and (10) are identical to thoseobtained for pure materials in the sharp interfacelimit [16,24].The phase-field mobility, M in Eq. (5), can be

determined using the chemical rate theory forsolidification [30–33,35]. Through the same ap-proach introduced in appendix of Ref. [24],the phase-field mobility at the thin-interface

limit is determined from the followingequation:

RTctot

V 0¼

sM�2

��ffiffiffiffiffiffi2w

p

xðcL;e1 ; . . . ; cL;en�1; cS;e1 ; . . . ; cS;en�1Þ; ð11Þ

where ctot ¼ 1=DþPn�1

i¼1;i2I ci; V 0 is the limitingvelocity at an infinite driving force andxðcL;ei ; . . . ; cS;ei ; . . .Þ is a temperature-dependentfunction defined by

xðcL;e1 ; . . . ; cL;en�1; cS;e1 ; . . . ; cS;en�1Þ

� �Xn�1j¼1

ðcL;ej � cS;ej ÞXn�1k¼1

ðcL;ek � cS;ek Þ

Z 0

1

Z 0

f0

Bejk

½1� hðfÞfð1� fÞ

dfh0ðf0Þdf0 ð12Þ

and Bejk is a component of the inverse matrix

of ~M at the equilibrium state. In the case of adilute binary alloy system, the relationshipbetween V 0 and the interface kinetic coefficient[34] reduces the relationship (11) to the existingrelationship between the interface kinetic coeffi-cient and the phase-field mobility [14,15,23]. In thesharp interface limit that the thickness of themoving solid/liquid interface vanishes, i.e.�=

ffiffiffiffiffiffi2w

p! 0; the phase-field mobility reduces

RTctot=V 0 ¼ s=ðM�2Þ; which is only valid for thecase that the inter-diffusion potential of eachsolute atom is constant across the interfacialregion.

3. Kinetics of the multicomponent alloy

solidification

The dilute solution limit is often useful in bothengineering applications and finding out funda-mentals. Here, it is shown that a steady-statesolution of the concentration profile across thediffuse interface can be obtained as a function ofthe interface velocity and that this solution reducesto the solution obtained in the classical sharp-interface model when the thickness of the movinginterface is much smaller than that for thediffusion boundary layer in front of the interface.Throughout this section, the following assump-tions are made: (i) the off-diagonal components ofdiffusivity matrix are neglected, (ii) each diagonaldiffusivity term, Dii; in both the interfacial regionand liquid is constant, and (iii) diffusivities in thesolid phase are negligible.In the dilute solution limit, the inter-diffusion

potential of a phase U is expressed as follows:

f UcU

i¼ AU

i þRT

DlnðXU

i Þ; (13)

where AUi is a constant for the ith component

element in phase U: XUi is defined as XU

i forinterstitial element and XU

i =XUn for substitutional

element in U phase, where XUi and XU

n represent

the mole fractions of the solute atom and solvent

atom, respectively, and XUi is defined by

cUi =ð1=DþPn�1

i¼1;i2I cUi Þ: Eq. (13) produces that

f LcLi

cLkwith iak is negligible in comparison with

f LcLi

cLiwhen both ith and kth components are

substitutional solute elements and that f LcLi

cLkwith

iak becomes zero when the ith or the kthcomponent is interstitial.Eq. (4) yields the following relationships for

both substitutional and interstitial solute at thelimit that all n � 1 compositions go to zero, i.e.XU

n ’ 1:

cSicLi

¼cS;ei

cL;ei

¼ kei ; (14)

where kei represents the equilibrium partition

coefficient of the ith alloy component.

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In a one-dimensional steady state with theinterfacial velocity, V ; the governing equation forconcentration is transformed to

d

dxf LcL

i¼ �

f cici

Dii

V ðci � cS;ii Þ; (15)

where cS;ik indicates the composition of the kthelement at the solid side (x ¼ �l) of the interface.At the limit where all n � 1 compositions vanish,we can get the following approximations:

d

dxf LcL

i¼ f LcL

icL

i

dcLidx

�RT

D1

cLi

dcLidx

; (16)

ci � ½1� ð1� kei ÞhðfÞc

Li (17)

and

f cici�

RT

D½1� ð1� kei ÞhðfÞc

Li

: (18)

Therefore, Eq. (15) becomes

dcLidx

þV

Dii

cLi ¼V

Dii

cS;ii

1� ð1� kei ÞhðfÞ

: (19)

Under the boundary condition cLi ¼ cSi =kei ¼

cS;ii =kei at x ¼ �l; the solution of this first-order

inhomogeneous equation is

cLi ¼cS;ii

kei

e�V ðxþlÞ=Dii þV

Dii

cS;ii e�Vx=Dii

Z x

�l

eVx0=Dii


dx0: ð20Þ

For the case that the thickness of the movinginterface is much smaller than that for thediffusion boundary layer in front of the interface,it can be easily shown that cLi ðxÞ at xbl convergesto the following solution obtained in the classicalsharp-interface model with the same diffusivity inliquid as Dii:

cLi ¼ cS;ii � cS;ii 1�1

kei

e�Vx=Dii : (21)

From Eqs. (17) and (20), the concentration profileciðxÞ across the interface is given by

cLi ¼ ½1� ð1� kei ÞhðfÞ

cS;ii

kei

e�V ðxþlÞ=Dii þV

Dii

cS;ii e�Vx=Dii

"

Zx

�l

eVx0=Dii


dx0

35: ð22Þ

Eq. (22) shows very interesting phenomenon, whichoccurs only in multicomponent alloy system. Therecan arise a regime of solidification, that can becharacterized by the absence of partitioning of theslow diffusing element. This type of transformation iscalled para-equilibrium, which was first introduced byHultgren in 1947 [38]. Under para-equilibrium, fastdiffusing solute atoms diffuse at an appreciable rate,but slow diffusing ones diffuse so slow that thegrowing phase inherits the same content of slowdiffusing solutes from the parent phase. To examinethis phenomenon, an ternary alloy system is selected,where the diagonal diffusivity terms and the equili-brium partition coefficient of both solute elementsare: DL

11 ¼ 2 10�9 m2=s; DL22 ¼ 2 10�8; ke

1 ¼

0:5; and ke2 ¼ 0:5: Concentration profiles are calcu-

lated from Eq. (22). Because the diffusion rate of thesecond solute is faster than that of the first one, thefirst element is expected to be trapped earlier than thesecond one as the interface velocity increases.Concentration profiles of the solute elements forvarious interface velocities are displayed in Fig. 1. Forincreasing interface velocity, both solute elementsshows decrease of partitioning and at 1m/s of theinterface velocity, the slow diffusing element istrapped while the fast diffusing one has appreciablepartitioning. Thus solidification with more than 1m/sof the migration rate shows para-equilibrium and ifthe interface velocity further increases, the fastdiffusing solute is expected to be also trapped.

4. Results and discussion

In this section, several numerical simulations forsolidification of a ternary alloy are presented. Fe—

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Fig. 1. Concentration profiles of the first and second solutes for various interface velocity. The equilibrium partition coefficients of

both solutes are the same and set at 0.5 and the diffusivity of the second solute atom is larger by order of one than that of the first one.


0.1 at% Mn—1.0 at% C alloy system, which hasnot only the substitutional solute atoms but alsointerstitial solute atoms, is selected for thisnumerical study. This alloy system solidifies to d-ferrite under the liquidus temperature 1793.5K.The equilibrium volume fraction of solid, f V ; andthe equilibrium concentrations of both solid andliquid phases are: f V ¼ 0:515; XL;e

Mn ¼ 1:172 10�3; XL;e

C ¼ 1:738 10�2; XS;eMn ¼ 8:389 10�4;

and XS;eC ¼ 3:091 10�3:With the numerical study

of one-dimensional isothermal solidification, thephase-field approach is shown to predict reason-ably well the equilibrium properties and thekinetics for a multicomponent alloy system. Somepreliminary results for the morphological evolu-tion of a two-dimensional dendrite are presented.

4.1. Equilibrium and kinetics in one-dimensional

isothermal solidification

To examine equilibrium properties and kineticsat the solidification of the selected ternary system,the computer simulation of the one-dimensionalisothermal solidification is performed. In thisstudy, the selected ternary alloy system is regardedas a dilute solution and under this assumption the

Eqs. (5) and (6) reduce into:

1

M

qfqt

¼ �2q2fqx2

þ h0ðfÞ

RT

Dln

XLn XS;e

n

X SnXL;e

n

� wg0ðfÞ;

(23)

qck

qt¼

qqx

Xn�1i¼1

Mki

qf ci

qx; (24)

f ci¼

RT

DðhðfÞ lnXS

i þ ð1� hðfÞÞ lnXLi Þ: (25)

Eqs. (23) and (24) are discretized by a second-order differencing scheme for the spatial deriva-tives and a simple explicit Euler scheme for thetime derivative. The no-flux boundary condition,rnf ¼ rnci ¼ 0; is imposed at both ends of thecomputation domain.In order to test the equilibrium properties at

solidification, solidification process from a smallsolid seed is simulated. The solid seed has the samecomposition as the liquid and is located at one endof the computation domain. The material proper-ties used in simulations are: s ¼ 0:4 J=m2; 2l ¼

600 nm and a ¼ 2:94: For the test of the equili-brium properties, the selection of the input

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Fig. 2. Concentration profiles for (a) Mn and (b) C in one-

dimensional Fe—0.1 at% Mn—1.0 at% C alloy. The times are

0; 1 10�2; 2 10�2; 3 10�2; 5 10�2; 0.24 s.


parameters related with kinetics are not important.Here, V0 is set to 0.5m/s and only diagonal termsof the solute diffusivity matrix are considered andare set to 2 10�8 m2=s for both phases. Note thatthe diffusivity matrix of each component for eachphase could be obtained in a general way [36,37].In computer simulation, the total grid number

in the system is fixed at 200 and grid spacing is setto be 100 nm such that an interfacial region couldbe covered with six grid spacings. The timeadvancing step, Dt; is set at ðDxÞ2=5DL

CC; wellbelow the upper numerical instability limit,ðDxÞ2=2DL

CC:Time-dependent concentration profiles for Mn

and C are displayed in Fig. 2(a) and (b),respectively. The sharp concentration profile atthe initial stage can be explained with transienteffect due to the initial uniform concentrationdistribution over whole computation domain.After solidification proceed, the solute concentra-tions in both phases reaches equilibrium at about0.24 s. At equilibrium, the predicted concentra-tions are 0.0837 at% Mn and 0.3091 at% C in thesolid phase with a volume fraction of 0.517, and0.1170 at% Mn and 1.7385 at% C in the liquidphase: they are in excellent agreement with thegiven thermodynamic data.In order to check the ability of the model to

consider the kinetics of the solidification, thenumerical solution by the phase-field model for asteady-state solidification in one-dimension iscompared with exact one obtained with theclassical sharp interface model. In the case thatdiffusion occurs in liquid only and there is a solutesink in the liquid phase away from the solid–liquidinterface such that the interface moves withconstant velocity V ; the concentration profileobtained with the classical sharp-interface modelis (24):

cLi ðxÞ ¼ c1i þ c1ið1� ke

i Þðe�Vx=DL

ii � e�Vx�=DLii Þ

1� ð1� kei Þð1� e�Vx�=DL

ii Þ;

(26)

where x� is the prescribed distance between theinterface and the solute sink in liquid and cL;ii c1iare the concentration of the ith component at theliquid side of the interface and the concentration

far from the interface, respectively. Eq. (26) hasthe unique value at any position if V and x� aregiven.The selected alloy system at 1780K is tested for

comparison between the exact solution and thephase-field model. The following additional mate-rial properties are used: DS

MnMn ¼ 1:0

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Fig. 3. Interface concentration versus solute sink distance x�:The top figure stands for the variation of Mn concentration

while the bottom represents C concentration in the liquid phase

at the interface. The solid circles represent the phase-field

results and the solid curves are from the analytic solution of Eq.

(26). The unit of each concentration is mole fraction.


10�14 m2=s; DSCC ¼ 1:0 10�9; DL

MnMn ¼

5:0 10�9; DLCC ¼ 2:0 10�8; ke

Mn ¼ 0:716; andkeC ¼ 0:178: The interface thickness, 2l is set at6 nm and grid size of 1 nm is used. The phase-fieldmobility is assumed to be 0.5. For the phase-fieldcalculation, a long-time behavior of its fulldynamics is closely examined to determine asteady-state velocity, V ; which is in turn used forthe analytical solution of Eq. (26). In Fig. 3, theconcentrations for Mn and C at the liquid side ofthe interface are plotted as a function of x�: Anexcellent agreement is shown for the two resultswith less than 1% error.

4.2. Two-dimensional dendritic growth of a ternary

alloy

The phase-field model is here extended to a two-dimensional solidification problem. In order to seethe effect of anisotropic interfacial energy, anenergy form adopted by Kobayashi [8] is used:

� ¼ �Z ¼ �½1þ ge cosð4yÞ; (27)

where � and ge are constants and y ¼

tan�1ðfy=fxÞ: Consequently, the system has afour-fold symmetry and its governing equationsfor a two-dimensional dendritic growth become

1

M

qfqt

¼ �2ZZ0½sinð2yÞðfyy � fxxÞ þ 2 cosð2yÞfxy

� ð1=2Þ�2½Z02 þ ZZ00

½2 sinð2yÞfxy � r2f� cosð2yÞ

ðfyy � fxxÞ þ �2Z2r2f� f f ð28Þ

and

qck

qt¼ r �

Xn�1i¼1

Mkirf ci; (29)

where Z0 ¼ dZ=dy and Z00 ¼ d2Z=dy2:Using the Eqs. (28) and (29) two-dimensional

computer simulations were performed for theisothermal solidification of the selected ternaryalloy system (Fe—0.1 at% Mn—1.0 at% C alloysystem) at 1770 and 1780K. In computer simula-tion, this alloy system is regarded as a dilutesolution. The interface thickness, 2l; is set at1200 nm with a grid size equal to 200 nm, theinterfacial energy, s; is equal to 0.4 J/m2. Other

ancillary input data are: ge ¼ 0:06; DSMnMn ¼ 1:0

10�14 m2=s; DSCC ¼ 1:0 10�9; DL

MnMn ¼

5:0 10�9; and DLCC ¼ 2:0 10�8: As before, the

off-diagonal diffusivity terms are assumed to be

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Fig. 4. (a) Mn concentration and (b) C concentration profiles

in the dendritic growth of the Fe—0.1 at% Mn—1.0 at% C

alloy at 1770K and at the evolution time of 1:25 10�4 s:


zero. When the phase-field mobility is determined,it is assumed that V 0 is independent of thetemperature and is set to 5.5m/s. The phase-fieldmobilities, which are determined by the Eq. (11),are 0.16 at 1770K and 0.0582 at 1780K. Duringthe numerical calculation for Eq. (28), a stochasticnoise is added at the interface region in order toinclude fluctuations at the interface, which gaverise to the initiation of the morphological in-stability [39].Equiaxed dendrite shape at 1770K is pictured at

the evolution time of 1:25 10�4 s in Fig. 4.Taking advantage of the four-fold morphologicalsymmetry, only a quarter of the entire dendrite ischosen as the computational cell, and a solid seedis placed at the corner of the bottom-left, that is atthe origin. Therefore, the dendrite picture in Fig. 4is that obtained by reflecting the calculated resultsabout the y-axis. No-flux conditions are used atthe right and top boundaries of the computationalcell. The CPU hours are about 48 on an IntelPentium IV—2GHz computer.The growth pattern in Fig. 4 shows many

features consistent with those observed in thework of Warren and Boettinger [39]. The primarystalk has a low concentration, but the regionsbetween the secondary arms have the highest.There is a tendency for the sidebranches to growinitially not perpendicular to the primary stalk,but then later to grow into the perpendiculardirection. Behind the dendrite tip, the roots ofsecondary arms are usually narrower than theirmidriffs. These features are commonly observed inreal dendrites.A close examination reveals that spines of some

secondary branches show a concentration as lowas that along the spine of the primary stalk. Such alow concentration along the spine of a primarystalk appears due to capillary effect at the growingtip and it is natural to see such a spine even in asecondary dendrite arm. In the ternary Fe–Mn–Csystem, the carbon diffusivity in the solid phase isalmost as fast as the diffusivity of Mn in the liquidphase. Consequently, carbon homogenization pro-cess is expected to occur even in the solid phase.Thus it is not a surprise to observe that in Fig.4(a), the Mn concentration along the primaryspine, i.e. along the vertical line at the center,

remains low all the way to the original seed place,while the carbon concentration along the primaryspine in Fig. 4(b) is already homogenized.Fig. 5 shows the equiaxed dendrite shape at

1780K with that at 1770K. The spatial distribu-tions of Mn concentration at the same evolutiontime for 1780 and 1770K are presented in Fig. 5(a)and (b), respectively. In comparison with the

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Fig. 5. The spatial distributions of Mn concentration at the same evolution time of 1:25 10�4 s for (a) 1780K and (b) 1770K. Later

stage of the growth of the dendrite at 1780K is shown in (c) and (d), which represents the concentration profile of Mn and C at

1:0 10�3 s; respectively.


dendrite at 1770K, the velocity of the solid–liquidinterface at 1780K is slow and the tip radius of thedendrite is larger due to the low undercooling. Dueto the low velocity of the solid–liquid interface, thepartitioning of Mn atoms becomes clearly and thisis consistent with the results in Section 3. Laterstage of the growth of the dendrite is shown in Fig.5(c) and (d), which represents the concentrationprofile of Mn and C at 1:0 10�3 s; respectively.From Fig. 5(c), the homogenization of Mn atomsis progressed conspicuously in solid phase, whichcan be explained by partitioning due to lowinterface velocity and enough time to diffuse insolid phase. The carbon concentration is alreadyhomogenized as a matter of course.Until now our numerical simulations were

presented only for a dilute ternary alloy. Innumerical simulations of non-dilute multicompo-nent alloys, there are some complexities related to

solve the condition (4) of the equal chemicalpotentials in solving a set of Eqs. (5), (6), (3), and(4). Contrary to the dilute solution approximation,there is no analytic expression to solve theconstraint equations (4) in general non-dilutemulticomponent alloys nor even in non-dilutebinary alloys. In binary alloy systems, Kim et al.have proposed a useful and efficient numericalmethod to solve it [23] but it is very difficult toextend their method to general multicomponentalloy systems (for ternary alloys, it is possible touse their method by making the table of composi-tions cSi and cLi of the solid and liquid phases asfunctions of the chemical potentials f ScS

i¼ f LcL

i¼

f ci; but for non-dilute multicomponent alloys with

more than three components, this method is notavailable.). We have used multi-dimensional

Newton–Raphson method for nonlinear systems of

equations (MDNR) [40] to solve the constraint

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equations (4). In general, as the free energyfunction is well defined (positive definite) one ofthe solute compositions, the convergence rate ofMDNR is not too bad. We have simulated thedendritic growth of the same system as shown inFig. 4 with MDNR. The calculation time wasincreased by � 350%; when compared with thesimulation using the dilute solution approximation(with MDNR, it takes one week to calculate thesystem shown in Fig. 4 on Intel Pentium IV—2GHz. Recently, as the computing performance ismore advanced, it is acceptable). AlthoughMDNR is time-consuming method, we can con-sider it as one of the numerical methods to solvethe constraint equations (4) of the equal chemicalpotentials for non-dilute alloys.

5. Conclusions

A phase-field model is presented for solidifica-tion of a multicomponent alloy system. The modelcould be applicable to any alloy systems with bothsubstitutional and interstitial elements by using thenumber of moles as concentration variable and itrelieves the limit of interface thickness due to thecontribution of the chemical free energy density inthe interfacial region which seriously restricts thecalculation domain size [24]. One-dimensionalnumerical calculations for solidification of anFe–Mn–C alloy system demonstrate that themodel could provide a good description for theequilibrium and kinetic properties for isothermalternary solidification. It is also demonstrated thatthe model could be extended to simulate theevolution of large-scale dendritic microstructuresand to predict the pattern of the solute distributionboth in the solid and in the liquid phase.

Acknowledgements

This work was supported by the internalresearch fund 2004 of Kookmin University inKorea.

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Phase-field model for multicomponent alloy solidification

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