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Journal of Crystal Growth 309 (2007) 65–69

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Phase-field simulatizon of microstructure development involvingnucleation and crystallographic orientations in alloy solidification

Junjie Li�, Jincheng Wang, Gencang Yang

State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, PR China

Received 27 March 2007; received in revised form 17 August 2007; accepted 26 August 2007

Communicated by G.B. McFadden

Available online 2 September 2007

Abstract

A proper treatment of nucleation in phase-field simulations is developed, in which the calculation of nucleation kinetics in binary melts

is taken into account and the free energy change of nucleation is obtained by maximizing it relative to the composition of the nuclei.

Using the phase-field model for polycrystalline solidification coupled with the nucleation strategy, the typical solidification

microstructures encountered in castings are simulated. The selection of columnar grains and the columnar-to-equiaxed transition

(CET) are clearly shown. The results also indicate that the columnar zone length and the equiaxed grain size increase with a decrease in

cooling rate, which results from the competition between nucleation and growth.

r 2007 Elsevier B.V. All rights reserved.

PACS: 64.70.Dv; 81.30.Fb; 64.60.Qb

Keywords: A1. Phase-field model; A1. Solidification; A1. Nucleation; B1. Binary alloy

1. Introduction

The microstructure formed during solidification plays anessential role in determining the final quality of the castingproduct. Therefore, in recent years a lot of studies havebeen undertaken to simulate the solidification microstruc-ture. Using the cellular automaton (CA) model, Rappazand Gandin [1,2] gave the first simulation results, whichhad a physical basis and resembled very closely observa-tions in real micrographic cross-sections. However, in theirpapers the solute diffusion equation is not considered, andthe growth velocity of a dendrite tip is predefined by theanalytical model before simulation. Some modified CAmodels [3–5] that directly solve the transport equations atthe solid–liquid interface have been developed to simulatedendritic growth patterns in solidification. In this way, arealistic-looking dendritic microstructure is produced andmuch insight into the solidification process is obtained. Butin these CA models the effects of crystallographic orienta-

e front matter r 2007 Elsevier B.V. All rights reserved.

rysgro.2007.08.025

ing author. Tel.: +8629 88460650; fax: +86 29 88492374.

ess: lijunjie@mail.nwpu.edu.cn (J. Li).

tion are considered in a relatively arbitrary way and sufferfrom the lack of a physical basis. The phase-field method isanother powerful tool for simulating the complex solidifi-cation microstructure. It has been widely used to study theproblem of the growth of a single dendrite in undercoolingmelts [6–8] and interface morphology in directionalsolidification [9–11]. Not only qualitative but also quanti-tative results have been obtained. With the advance ofcomputational power, the phase-field method has also beenextended to simulate multigrain structures [12–14].Nucleation and arbitrary crystallographic orientations

are two important factors for the simulation of multigrainstructure. Besides the classical nucleation model, peoplehave developed many sophisticated methods [15–17] tomodel homogeneous and heterogeneous nucleation. Re-cently, phase-field-type models have been proposed tohandle heterogeneous nucleation with a diffusive interface[18–20]. Despite this inventory, in the simulation of theevolution of solidification microstructures, very simpletreatments of nucleation are still widely used due to thecomplexity of the problem and the restriction of computa-tional power. Some studies [14,21] use a single-valued

ARTICLE IN PRESS

LS

ALμ

BLμ

BSμ

BLμ

VGΔ

cB

cS cL

G

Fig. 1. Schematic Gibbs free-energy composition diagram. The free-

energy change DGV associated with forming a small nucleus of

composition cS in the liquid of composition cL is obtained by the parallel

tangent construction.

J. Li et al. / Journal of Crystal Growth 309 (2007) 65–6966

nucleation supercooling which is relatively arbitrary, andsome [1–3] use a Gaussian distribution of grain initiationevents as a function of supercooling. Since the parametersof the distribution cannot be directly obtained theoreticallyor experimentally, they are just adjusted to force agreementbetween prediction and experiments. These two kinds ofmethods cannot reflect the stochastic nature of nucleationand cannot reproduce classical nucleation theory (CNT)either.

In this paper, stochastic nucleation is taken into accountthrough the Poisson seeding algorithm proposed bySimmons and co-workers [22,23] and a kinetic calculationfor binary melts is performed based on the CNT. Thisapproach to the modeling of nucleation has been used tosimulate the microstructural development in solid-statetransformations [23,24]. Here with a proper modificationfor the calculation of free energy change of nucleation,which depends on both temperature and concentration,this nucleation strategy is incorporated into a phase-fieldmodel for polycrystalline solidification in binary alloys.Typical solidification microstructures encountered in cast-ings are reproduced by the phase-field simulation.

2. Methods

2.1. Nucleation strategy

Fundamentally, nucleation is a statistical process.Poisson statistic is a proper treatment of the stochasticnature of nucleation. Accordingly, Simmons et al. [22]treated nucleation as a fully localized and probabilisticevent, and proposed that the nucleation probability in asufficiently small time interval Dt and cell spacing v can beapproximated by unity minus the ‘‘zero event probability’’of a Poisson distribution: Pn ¼ 1� exp ð�IvDtÞ. Thenucleation rate I can be obtained from any general modelor even experimental data. Adopting the framework of theCNT, we utilize the traditional rate equations for steady-state nucleation in undercooled liquid

I ¼ I0 exp �16ps3f ðyÞ

3kTðDGVÞ2

� �, (1)

where I0 is a prefactor determined by the jump frequencyacross the interface, I0 � 1039�1 m�3 s�1 for volumenucleation and I0 � 1031�1 m�2 s�1 for nucleation onsubstrate surface, s is the solid–liquid interface energy, k

is Boltzmann’s constant, f ðyÞ ¼ ð2� 3 cos yþ cos3 yÞ=4with y as the contact angle, and DGV the Gibbs free energydifference between the melt and the crystal. The formulaDGV ¼ DH fDT=Tm � DCp½DT � T lnðTm=TÞ� is oftenused to calculate DGV when the nuclei have the samecomposition as the liquid, which is only valid for puresubstances but not for alloys. Thompson and Spaepen [25]suggested that the composition of the nuclei should bedetermined by maximizing DGV. The maximum of DGV

is achieved when the chemical potential change of

components A and B are equal as depicted in Fig. 1, so that

DGV ¼ DmA ¼ DmB. (2)

For general solutions, the chemical potential of compo-nent A or B in the liquid or solid can be expressed asfollows:

mLA ¼ GL � cLqGL

qcL, (3)

mLB ¼ GL þ ð1� cLÞqGL

qcL, (4)

mSA ¼ GS � cSqGS

qcS, (5)

mSB ¼ GS þ ð1� cSÞqGS

qcS. (6)

Therefore

DmA ¼ mLA � mSA ¼ GL � GS þ cSqGS

qcS� cL

qGL

qcL, (7)

DmB ¼ mLB � mSB ¼ GL � GS þ ð1� cLÞqGL

qcL� ð1� cSÞ

qGS

qcS.

(8)

Combining Eq. (2) gives

qGL

qcL¼

qGS

qcS. (9)

For an actual calculation of DGV, it is necessary to usesome solution models and the thermodynamic database toobtain the molar-free energy of the solid phase (GS) andliquid phase (GL). These can be expressed as

GS ¼ ð1� cSÞoGS

A þ cSoGS

B þ RTcS ln cS

þ RTð1� cSÞ ln ð1� cSÞ þ cSð1� cSÞISAB, ð10Þ

ARTICLE IN PRESSJ. Li et al. / Journal of Crystal Growth 309 (2007) 65–69 67

GL ¼ ð1� cLÞoGL

A þ cLoGL

B þ RTcL ln cL

þ RTð1� cLÞ lnð1� cLÞ þ cLð1� cLÞILAB, ð11Þ

where ISAB and ILAB are interaction parameters. Dependingon the values of these parameters, Eqs. (10) and (11) canrepresent ideal solution model, regular solution model orsub-regular solution model. Eqs. (8)–(11) can be solvedtogether to calculate cs and DGV. It should be mentionedthat the free energy of the solid and liquid phases are alsoincluded in our phase-field calculation with the same formsof Eqs. (10) and (11), so they are used directly in nucleationkinetic calculation.

It should be noted that the composition dependence ofthe interfacial free energy has been neglected here forsimplicity. In addition, a refined model could be obtainedby combining the approach of Simmons et al. [22] withnucleation barrier heights 16ps3f ðyÞ=ð3DG2

VÞ evaluatedfrom the phase-field theory itself (as described inRef.[12]). This will be done in the future.

Table 1

Thermodynamic properties of the Ni–Cu binary system (J/mol)

oGNiL 11235.527+108.457T�22.096T lnT�0.0048407T2

�3.82318� 10�21T7

oGCuL

�46.545+173.881484T�31.38T InToGNi

S�5179.159+117.854T�22.096T InT�0.0048407T2

oGCuS

�13542.026+183.803828T�31.38T InT+3.642� 1029T�9

LL0 11760+1.084T

LS0 8365.65+2.80242T

2.2. Phase-field model

Based on the work of Kim et al. [26], Kobayashiet al. [27] and Granasy et al. [28], we have developeda phase-field model for polycrystalline solidificationin binary alloys [29]; the free energy functional is postu-lated as

F ¼

Z�2

2jrfj2 þ f ori þ f ðc;fÞ

� �dV , (12)

where the orientational contribution f ori ¼ HhðfÞjryjrepresents the excess free energy due to inhomogeneitiesin crystal orientation in space. y (a normalized orienta-tional angle) is the non-conserved orientational fieldvariable, which is random in the liquid, but has a definitevalue between 0 and 1 in the crystal that determinescrystal orientation in the reference frame. The parameter H

is independent of c and quantifies the energetic cost ofmisorientation [30]. h(f) varies smoothly from 0 to 1as f changes from the liquid to the solid. As a result, theexcess energy acts in the solid only and is proportional tojryj. The local free energy density f(c, f) depends on thephase-field variable f and concentration field c. It takes theform

f ðc;fÞ ¼ hðfÞf SðcSÞ þ ½1� hðfÞ�f L

ðcLÞ þ wgðfÞ, (13)

where f SðcSÞ and f L

ðcLÞ are the free energy densities of thesolid and liquid phase. hðfÞ ¼ f2

ð3� 2fÞ and gðfÞ ¼f2ð1� fÞ2. c is determined to be the fraction-weighted

average value of the solid concentration cS and the liquidconcentration cL, c ¼ hðfÞcS þ ½1� hðfÞ�cL. In the inter-face region the chemical potentials of solid f S

cS½cSðx; tÞ�

and liquid f LcL½cLðx; tÞ� are assumed to be equal. Using

the standard approach of irreversible thermodynamics,the equations of motion for the three fields can be

obtained as

1

Mf

qfqt¼ rð�2rfÞ �Hh0ðfÞjrfj � wg0ðfÞ

þ h0ðfÞ½f LðcLÞ � f S

ðcSÞ � ðcL � cSÞfLcLðcLÞ�, ð14Þ

1

My

qyqt¼ Hr hðfÞ

ryjryj

� �, (15)

qc

qt¼ r½DðfÞrc� þ r½DðfÞh0ðfÞðcL � cSÞrf�. (16)

3. Results and discussion

Solidification process of Ni-0.396mol%Cu alloy issimulated. The free-energy densities of solid and liquidare constructed by the regular solution models:

f SVm ¼ ð1� cSÞoGS

Ni þ cSoGS

Cu þ RTcS ln cS

þ RTð1� cSÞ lnð1� cSÞ þ cSð1� cSÞL0S, ð17Þ

f LVm ¼ ð1� cLÞoGL

Ni þ cLoGL

Cu þ RTcL ln cL

þ RTð1� cLÞ lnð1� cLÞ þ cLð1� cLÞL0L. ð18Þ

The thermodynamic data from the Al-v3 database ofThermo-Calc is listed in Table 1. These solute models andthermodynamic data will be used both in the calculation ofnucleation rate and phase-field equation. The calculation isperformed on a 2000� 2000 foursquare grid, and the gridsize is 5.0� 10�8m. The temperature of the system isassumed to be uniform and decreases with a constantcooling rate. Twenty nuclei are given at the left side of thesystem at the beginning, and they will grow into theundercooled liquid. The heterogeneous nucleation in thebulk liquid in the cooling process is simulated accordingto the nucleation model given above. We set I0 ¼

1030 m�2 s�1 and f ðyÞ ¼ 0:0015. The nucleation probabilityPn is calculated at each liquid node at each time stepthroughout the simulation. Meanwhile a uniform randomvariable is generated. If it is less than Pn, then the node istransformed; otherwise it is not.The final solidification structures for four different

cooling rates, _T ¼ 30; 75; 150 and 300 K=s, are shown inFig. 2. The initial temperature is 1580K for all the fourcases. The initial grains on the boundary of the left sidehave different crystallographic orientations, so they willimpinge on each other as they grow forward. The dendrites

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Fig. 2. Simulated microstructure for a Ni–0.369mol% Cu alloy solidified

under different cooling rates: (a) 30, (b) 75, (c) 150, (d) 300K/s.

1575 1576 1577 1578 1579 1580

0.0

4.0x105

8.0x105

1.2x106

2

4

6

8

Nu

cle

atio

n r

ate

Temperature (K)

Ve

locity (

mm

/s)

Fig. 3. The dependence of nucleation rate and dendrite tip velocity on the

temperature.

J. Li et al. / Journal of Crystal Growth 309 (2007) 65–6968

extend faster into the bulk liquid when their /0 0 1Scrystallographic orientation is perpendicular to the left-sideboundary, and the growth in other directions are sup-pressed. Therefore a columnar dendritic structure forms.This selection mechanism in the system with a uniformtemperature is not really based on a minimum under-cooling criterion, but rather on a ‘‘minimum travel’’criterion proposed by Rappaz and Gandin [1]. When thecooling rate is large enough(see Fig. 2(b)–(d)), nucleationin the bulk liquid can be observed in our calculationdomain. These newly nucleated grains can stop the growthof columnar dendrites and result in a columnar-to-equiaxed transition (CET).

It can be seen that the average length of the columnargrain, L, decreases with the increase in the cooling rate _T .L can be calculated as follows:

L ¼1

_T

Z Tnuc

T int

vðTÞdT , (19)

where Tint is the initial temperature at which the columnargrain nucleates; it is same for all the four cases in Fig. 2.Tnuc is the nucleation temperature of the equiaxed grain,which is usually defined as

1 ¼ V

Z Tnuc

TL

1

_TIðTÞ dT , (20)

where V is the volume of the liquid and TL is the liquidustemperature. Through calculation we found that thenucleation temperature Tnuc for different cooling rates isapproximately equal and only decreases 0.6K when _Tincreases from 30 to 300K/s. Therefore, according to

Eq. (19), the length of the columnar grain is mainly decidedby the cooling rate, and is inversely proportional to thecooling rate just as shown in Fig. 2. This behavior is inqualitative agreement with experimental data for binaryAl alloys [31]. For the simplifying assumptions used inour model (see the last paragraph in this section), althoughthe calculated results cannot be compared with theexperiment quantitatively, the basic mechanism of thebehavior in our simulation is the same as that mentioned inthe experiment [31].The average sizes of the equiaxed grains in Fig. 2(b)–(d)

are estimated at 1.81, 10.8 and 8.8 mm, respectively, whichshow a decrease with increased cooling rate. The averagegrain size is determined by the nucleation rate and growthvelocity. For a large nucleation rate, the number of thegrains will be large and the size of each grain will be small;however, at a much higher growth velocity it will be just thereverse, so the final grain size is the result of thecompetition between nucleation and growth. The dendritetip velocities at different constant temperatures (from 1580to 1575K at intervals of 1K) are evaluated from 6isothermal simulations. The variation in the nucleationrate and the dendrite tip velocity with the temperature areillustrated in Fig. 3. They all increase with the decrease intemperature, but the nucleation rate rises exponentially,while the growth rate rises linearly in the undercoolingrange concerned. It can be found that the variance ratio ofthe nucleation rate below the nucleation temperature,which approximately equals 1576K, is very large, whilethe change of growth velocity is relatively slow. There-fore, with the decrease in the temperature the effect ofincreasing nucleation rate will overwhelm the effect ofincreasing growth velocity. Moreover, the nucleation ratewill be larger when the decrease of temperature is faster.So a refined grain structure will form with increasedcooling rate.Finally, it has to be noted that in order to observe the

CET within our calculation domain, the cooling rate hasbeen enhanced compared with the realistic value in usual

ARTICLE IN PRESSJ. Li et al. / Journal of Crystal Growth 309 (2007) 65–69 69

casting process. The uniform temperature assumption usedhere is only valid if the domain of interest is much smallerthan the thermal diffusion length. A more quantitativesimulation must be carried on a scale comparable to thetypical experiments, and the temperature evolution must bealso considered. This will require additional computationalresources.

4. Conclusions

In conclusion, the calculation of nucleation kinetics inbinary alloys, which depends on both temperature andconcentration, is incorporated into a phase-field simula-tion. The free energy change of nucleation is obtained bymaximizing it relative to the composition of the nuclei.Using the nucleation strategy and the phase-field model forpolycrystalline solidification in binary alloys, in which thearbitrary crystallographic orientation is also involved,some realistic-looking solidification microstructures areproduced. The results also indicate that the columnar zonelength and the equiaxed grain size increase with thedecrease in cooling rate, which results from the competitionbetween nucleation and growth.

Acknowledgments

The authors acknowledge the support of the NationalNatural Science Foundation of China (Grant no.50401013) and the Doctorate Foundation of NorthwesternPolytechnical University.

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