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Journal of Crystal Growth 311 (2009) 1217–1222

Contents lists available at ScienceDirect

Journal of Crystal Growth

0022-02

doi:10.1

�Corr

E-m

journal homepage: www.elsevier.com/locate/jcrysgro

Investigation into microsegregation during solidification of a binary alloy byphase-field simulations

Junjie Li, Jincheng Wang �, Gencang Yang

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

a r t i c l e i n f o

Article history:

Received 18 October 2008

Accepted 23 November 2008

Communicated by M. Rettenmayrvarious back diffusion conditions. Under the condition of no back diffusion, it is found that for the region

without second dendrite arms the simulation result agrees well with the Scheil equation, but for the

Available online 3 December 2008

PACS:

64.70.D�

64.70.kd

81.30.Fb

64.60.Bd

Keywords:

A1. Microsegregation

A1. Phase-field model

A1. Solidification

A2. Dendritic growth

48/$ - see front matter & 2008 Elsevier B.V. A

016/j.jcrysgro.2008.11.078

esponding author. Tel.: +86 29 88460650; fax

ail address: jchwang@nwpu.edu.cn (J. Wang).

a b s t r a c t

Microsegregation within the columnar dendritic array is analyzed by performing a two-dimensional

phase-field simulation. The influence of microstructure morphology on microsegregation is studied for

region with well developed second dendrite arms there is a severe deviation. This deviation is attributed

to the dendrtic coarsening and the inhomogeneity of interdendritic liquid concentration caused by

various interface curvatures. Under the condition of moderate back diffusion, it is found that the effect

of dendritic morphology on microsegregation can be accounted by enhancing the Fourier number which

characterizes the solid-state diffusion. However, this effect decreases with enhancing the back diffusion

of the system.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The description of microsegregation is a key step towards acomplete understanding of the solidification process. There aremany models of microsegregation in the literature [1–11]. Thevarious models differ mainly in the way in which the solute issupposed to be distributed in the solid. Under the limitingcondition of complete diffusion or no diffusion in the solid,microsegregation can be treated by the well-known lever rule orthe Scheil equation. For the case of moderate back diffusion in thesolid some analytical models [1–4] have been derived afterintroducing approximations for the diffusion behavior in thesolid. To obtain more exact solutions for the solid-state diffusion,numerical models [5,6] have been developed. Available in theliterature are also a number of improved models [7–11] takingaccount of the dendrite arm coarsening and the dendrite tipundercooling.

In all the models listed above the grain morphology issimplified to one-dimensional (1D). However, in principle, micro-segregation has a close relationship with microstructures, so amore exact treatment of microsegregation should be coupled with

ll rights reserved.

: +86 29 88491484.

the complex solidification structures. Recently, Du and Jacot [12]have made the first step to study the effect of grain morphologyon microsegregation by using the pseudo-front tracking method[13,14]. They found that the evolution of the solid/liquid interfacein two-dimensional (2D) calculations can not be well approxi-mated by the 1D assumption. Therefore, back diffusion which isclosed related with the area of solid/liquid interface cannot bewell described by traditional microsegregation models. Du et al.[15] also conducted 2D simulations to explain the decrease ofnon-equilibrium eutectic fraction with increasing the cooling rate.They proposed that the difference in coarsening kinetics at variouscooling rates should account for this decrease. However, due tothe complication of the problem it is difficult to calculate thecoarsening exponent directly from their simulations or make aquantitative comparison with analytical microsegregation modelsaccounting for the coarsening effect.

The phase-field method has emerged as a powerful tool forsimulating the complex solidification microstructure. In the morerecent history, Karma [16] has developed a quantitative phase-field model for binary alloys. Using Karma’s approach, Kim [17]extended the quantitative model to multicomponent systemswith arbitrary thermodynamic properties. Provatas et al. [18]performed the first simulation of dendritic growth at theexperimental parameters. Bottger et al. [19] simulated the equiaxedsolidification in technical alloys by coupling the phase-field model

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J. Li et al. / Journal of Crystal Growth 311 (2009) 1217–12221218

with the thermodynamic data base. Bedillo and Beckermann [20]applied the quantitative phase-field model to simulate thecolumar-to-equiaxed transition in directional solidification.Granasy et al. [21] performed simulations to yield a variety ofpolycrystalline growth morphologies. Folch and Plapp [22] con-ducted quantitative simulations on triple junction motion using aeutectic phase-field model. For reviews more recent progress canbe found in Refs. [23–25].

In phase-field simulations, microstructural evolution andconcentration field can be explicitly tracked in a physical manner,which results in a convenience to analyze microsegregation. Inthis paper a 2D phase-field model for polycrystalline solidification[26] is employed to study microsegregation within the columnardendritc array. As mentioned above, the effect of the 2D grainmorphology on microsegregation has been analyzed by Du andJacot [12], but neither have they considered the condition of noback diffusion, nor have given a regularity of the effect fordifferent extents of back diffusion. So our concern here is whetherthe complex solidification structures still have a great influenceon microsegregation in the absence of back diffusion. Thesimulation results are compared with the analytical microsegre-gation models based on 1D simplification. We also study theinfluence of microstructure morphology on microsegregation fordifferent extents of back diffusion.

2. Model description

Based on the work of Kim et al. [27], Kobayashi et al. [28] andGranasy et al. [29], we have developed a phase-field model forpolycrystalline solidification in binary alloys [26]. Evolutionof the phase field, orientation field and concentration field aredescribed by:

1

Mf

qfqt¼ rð�2rfÞ � Hh0ðfÞjryj �wg0ðfÞ

þh0ðfÞ½f LðcLÞ � f S

ðcSÞ � ðcL � cSÞfLcLðcLÞ� (1)

1

My

qyqt¼ Hr hðfÞ

ryry�� ��

" #(2)

qc

qt¼ r½DðfÞrc� þ r½DðfÞh0ðfÞðcL � cSÞrf�, (3)

where we defined the phase field f=1 at solid and f=0 at liquid,h(f)=f2(3�2f) and g(f)=f2(1�f)2, fs(cS) and fL(cL) are the freeenergy densities of the solid and liquid phase. The solutediffusivity is defined as D(f)=DSh(f)+(1�h(f))DL, DS and DL arethe diffusion coefficient in solid and liquid, respectively. c isdetermined to be the fraction-weighted average value of the solidconcentration and the liquid concentration, c=h(f)cS+[1�h(f)]cL.In the interface region the chemical potentials of solid f S

cS½cSðx; tÞ�

and liquid f LcL½cLðx; tÞ� are assumed to be equal. It should be noted

this treatment which is originally proposed by Kim [27] caneliminate the chemical potential jump at interface and allows theinterface to be stretched while maintaining the same surfaceenergy. y (a normalized orientation angle) is the non-conservedorientation field variable, which is random in the liquid but has adefinite value between 0 and 1 in the crystal that determinescrystal orientation in the reference frame. See Li et al., [26] formore details about this model.

The phase-field parameters of e and w are related to interfaceenergy, s, and interface width, 2l, where f changes from 0.1 to0.9, and the parameter, Mf is related to the kinetic coefficient, b, in

a thin interface limit [27]. They are given by

s ¼ �ffiffiffiffiwp

3ffiffiffi2p (4)

2l ¼ 2:2ffiffiffi2p �ffiffiffiffi

wp (5)

b ¼Vm

RT

me

1� k

sMf�2

��

Di

ffiffiffiffiffiffiffi2wp zðce

S; ceLÞ

" #(6)

zðceS; c

eLÞ ¼ f S

ccðceSÞf

Lccðc

eLÞðc

eL � ce

SÞ2

�

Z 1

0

hðf0Þ½1� hðf0Þ�

½1� hðf0Þ�fSccðc

eSÞ þ hðf0Þf

Lccðc

eL�

df0

f0ð1� f0Þ(7)

where Vm is molar volume, R is the gas constant, me is the slope ofthe liquidus line in the phase diagram, k is the equilibriumpartition coefficient, Di is the diffusivity at the interfacial regionand will be taken as the same value as DL. The orientation mobilityis assumed to vary proportionally to h(f) across the interface,My=My,L+h(f)[My,S�My,L], where My,L and My,S are the mobility inthe bulk liquid and solid phases. Since we are primarily interestedin polycrystalline solidification that takes place on a far shortertimescale than grain boundary relaxation, My,S is set so that grainrotation is negligible on the time scale of solidification, and My,L isset to be comparable to the phase-field mobility.

Solidification processes of the Ni–0.396 mol%Cu alloy underconstant cooling conditions are simulated. Material parametersused for computation are as follows: liquid diffusion coefficientDL=1�10�9 m2/s, solid diffusion coefficient Ds=0 m2/s for no backdiffusion condition and Ds=10�12, 5�10�12, 1�10�11, 5�10�11

and 1�10�10 m2/s for various extents of back diffusion conditions,interface energy s=0.35 J/m2, partition coefficient k=0.87, kineticcoefficient b=0 Ks/m. For the evolution equation of orientationfield H=0.1 J/m2, My,S=1�10�5Mf and My,L=5Mf are applied. Meshsize Dx=Dy=1.0�10�7 m, time step Dt=1.25�10�6 s, computa-tional domain size 2000�2000 in terms of grid number. Anexplicit finite difference scheme is employed to solve thegoverning equations of phase field and concentration. A fullyimplicit finite difference method is used to solve the evolutionequation of orientation field. The system temperature is set to beuniform and decreases from 1580 K with a constant cooling rate of75 K/s. As an initial condition, twenty nuclei are given at the leftside of the system, and they will grow into the undercooled melts.

3. Results and discussion

3.1. Under no back diffusion condition

The simulated microstructure at t=31.25 ms is shown in Fig. 1.As illustrated in this figure the system can be divided into threezones: the mushy zone (zone I), the dendrite tip zone (zone II) andthe liquid zone (zone III). The extent of liquid mixing is different inthe mushy zone and dendrite tip zone, which can be found fromthe evolution of concentration along the line a–a (see Fig. 2).During the formation of dendrite tip zone (t=31.25 ms) there is alarge concentration gradient in the liquid phase. Only after theformation of mushy zone (t=37.5, 62.5, 125 ms) the liquid can mixwell. Therefore in the whole solidification process the relationshipbetween the solid fraction and the liquid concentration cannot becorrectly described by general microsegregation models in whicha complete mixing of liquid is commonly assumed. As shown inFig. 3 apparently, the simulated results in the region A (the areawith x changing from 10 to 30mm) and region B (the area with x

changing from 100 to 130mm) seriously deviated from the Scheilequation.

ARTICLE IN PRESS

00

40

80

120

160

200

y (µ

m)

a

a

I II III

x (µm)40 80 120 160 200

Fig. 1. Growth morphology (composition map) and schematic representation of

three zones.

66

0.38

0.40

0.42

0.44

0.46

c (a

t%)

t = 31.25mst = 37.5mst = 62.5mst = 125ms

727068 74 76 78 80 82y (µm)

Fig. 2. Concentration evolution along the line a–a.

0.0

0.40

0.44

0.48

0.52

0.56Scheil equationsimulation result in region Asimulation result in region B

0.2 0.4 0.6 0.8 1.0

c l (a

t%)

fs

Fig. 3. Liquid concentration profiles as a function of the solid fraction for the

whole solidification process.

0.00

2

4

6

8

10

12

14

16

18

the staring pointdisscussed in our paper

for region Afor region B

fS

unde

rcoo

ling

(K)

0.2 0.4 0.6 0.8 1.0

Fig. 4. The undercooling evolution as a function of the solid fraction.

J. Li et al. / Journal of Crystal Growth 311 (2009) 1217–1222 1219

It should be noted that the high initial undercooling andcooling rate in our simulation will lead to non-equilibriumsolidification. This also makes the simulation results deviate fromthe analytical microsegregation model. However our analysesindicate that non-equilibrium solidification only happens duringthe initial stage of dendrite tip growth. The evolution of the totalundercooling as a function of the volume fraction of solid forregions A and B are shown in Fig. 4. It can be seen that at thebeginning of solidification there is a high undercooling. So non-equilibrium solidification occurs at this stage. With the progressof solidification the solute constituents are rejected into the liquidphase. When the volume fraction of solid is high, which meansthat the mushy zone has formed, the undercooling in regionsA and B will become very low. So the solidification in the mushyzone is near-equilibrium.

In order to describe the microsegregaion in the rapidsolidification process of dendritic or cellular array, Giovanolaand Kurz [8] proposed to choose two compatible functions for the

concentration vs. fraction relationship: one for the transient in theregion around the dendrite tips and the other for the regime ofcomplete liquid mixing of the interdendritic liquid. In thetransient regime Giovanola and Kurz [8] used a polynomial formto describe the relation of solid fraction with the solid composi-tion at the interface. A general dendrite growth model suitable fornon-equilibrium solidification is needed to ascertain the para-meter in the function for transition regime. In the completemixing regime the Scheil equation was employed by Giovanolaand Kurz [8] to describe the relation between the solid fractionand the liquid concentration. However, in this situation the initialstate should be cL=cx at fs=fx (instead of cL=c0 at fS=0 in the case ofstandard Schiel equation). The Scheil equation was modified as

cL ¼ cx 1�f S � f x

1� f x

� �ðk�1Þ

(8)

where fx and cx are the solid fraction and the liquid concentrationat the end of the transient regime.

Our concern here is the influence of microstructure morphol-ogy on microsegregation but not the effect of liquid concentrationgradient or undercooling on microsegregation. So based on theabove analyses only the near-equilibrium solidification in thecomplete mixing regime (i.e., after the formation of the mushyzone) is carefully analyzed in the following study. The startingpoints of our analysis for regions A and B are shown in Fig. 4.

ARTICLE IN PRESS

Fig. 6. Simulated microstructure evolution for region A (a) and region B (b).

0.00

0.05

0.10

0.15

0.20

0.25

Are

a Fr

actio

n

region Aregion B

J. Li et al. / Journal of Crystal Growth 311 (2009) 1217–12221220

During our simulation we observe the variation of undercoolingand liquid composition in a certain region (region A or B). Whenthe undercooling is relatively small and the concentrationgradient in the liquid almost disappear, we think that solidifica-tion in this region comes into the complete mixing regime. Wechose the solid fraction and the liquid concentration at this timeas the initial condition in the Scheil equation. Since the backdiffusion is absent in the simulation, if the evolution of the solid/liquid interface in our 2D simulation could be well approximatedby the 1D assumption, the Scheil equation (Eq. (8)) would be aproper description of the simulation results. The consistencebetween the Scheil equation and the simulation results are trulyobserved in region A (the area with x changing from 10 to 30mm),just as shown in Fig. 5. However, it can also be found in Fig. 5 thatthere is a large deviation between the Scheil equation and thesimulation results in region B (the area with x changing from 100to 130mm). The reason of the deviation in region B must link withthe microstructure morphology, because the undercoolings inregions B and A are approximately the same during the stage westudied.

The microstructure evolutions in regions A and B are shown inFig. 6. It can be seen that the second dendrite arms are welldeveloped in region B since the initial undercooling is high whendendritic growth begins here. Solidification in region B after theformation of mushy zone is accompanied by the coarsening ofsecond dendrite arms, which can influence microsegregation andmay account for the deviation between the simulation results andthe Scheil equation. Voller and Beckermann [11] and Mortensen[7] have proposed two analytical microsegregation modelsaccounting for the coarsening effect under zero back diffusionassumption. Following the way of modifying Scheil equation, theabove two models can be expressed as

cL ¼ cx2n½1� ðf s � f xÞ=ð1� f xÞ�

ð1þ2nÞk�1

½ðf s � f xÞ=ð1� f xÞ�2n

�

Z ðf s�f xÞ=ð1�f xÞ

0F2n�1

ð1�FÞ�ð1þ2nÞkdF (9)

f s ¼ 1� ð1� f xÞ1þ n

1� k

c½k=ð1�kÞ�L

ðcL � cxÞn

Z cL

cx

F½k=ð1�kÞ�ðF� cxÞ

ndF (10)

where n is the coarsening exponent which equals 13 in theory.

When n equals zero Eqs. (9) and (10) will reduce to the Scheil

0.76

0.44

0.46

0.48

0.50

0.52

0.54

0.56

c l (a

t%)

simulation results in region Asimulation results in region Bmodified Scheil Eq. (8)Eq. (9)Eq. (10)

0.80 0.84 0.88 0.92 0.96fs

Fig. 5. Liquid concentration profiles as a function of the solid fraction for the

complete mixing stage.

0.456cl (at%)

0.458 0.460 0.462 0.464 0.466

Fig. 7. Histograms of the liquid concentration for different regions at t=100ms.

equation. The relations between solid fraction and liquidconcentration according to Eqs. (9) and (10) for n ¼ 1

3 areillustrated in Fig. 5. The simulation results still severely deviatefrom these analytical microsegregation models. Another possibleinfluence factor is the inhomogeneity of interdendritic liquidconcentration. Although the liquid can mixed well in theinterdendritic region, various solid/liquid interface curvaturesinduced by the well developed second arms will change the liquidconcentration, which is just the driving force for coarsening. Theliquid concentration histograms for regions A and B at t=100 mspresented in Fig. 7 show that the peak in region B is much broaderthan that in region A, which means more inhomogenous for liquidconcentration in region B. However, this inhomogeneity of liquidconcentration is absent in the above analytical microsegregationmodels. So even the coarsening effect is considered, there is still alarge deviation between simulation results and analytical models.

ARTICLE IN PRESS

0.78

0.445

0.450

0.455

0.460

0.465

0.470

0.475

c L (a

t%)

0.450

0.455

0.460

0.465

0.470

0.475

c L (a

t%) DS = 5x10-11 m2/s

DS = 1x10-11 m2/s

DS = 5x10-12 m2/s

DS = 1x10-12 m2/s

DS = 1x10-10 m2/s

0.80 0.82 0.84 0.86 0.88 0.90 0.92fS

DS = 5x10-11 m2/s

DS = 1x10-11 m2/s

DS = 5x10-12 m2/s

DS = 1x10-12 m2/s

DS = 1x10-10 m2/s

J. Li et al. / Journal of Crystal Growth 311 (2009) 1217–1222 1221

3.2. Under various moderate back diffusion conditions

For the condition of moderate back diffusion we still only studythe near-equilibrium solidifications after the formation of mushyzone in regions A and B. In this situation the traditional Brody-Flemings model is a possible choice to describe microsegregation.Following the way of modifying Scheil equation, it can bemodified as

cL ¼ cx1� ð1� 2akÞf S

1� ð1� 2akÞf x

� �ððk�1Þ=ð1�2akÞÞ

(11)

where a is the Fourier number, which characterizes the solid-statediffusion. Some modifications have been proposed to get moreexact solutions to the back diffusion problem [2–4]. Since ourconcern here is the effect of the extent of back diffusion onmicrosegregaion in regions with different microstructures, thereis no need to calculate the back diffusion exactly in theory. Wewill take a as a fitting parameter and not use the moresophisticated back diffusion models [2–4]. It can be seen thatthe modified Brody-Flemings equation will reduce to the modifiedScheil equation with a=0, and reduce to the modified Lever rule,

cL ¼ cx1� ð1� kÞf x

1� ð1� kÞf S

(12)

with a=0.5.Firstly we study the situation for relatively small solid

diffusion coefficient, DS=1�10�12 m2/s. In this situation themicrostructure morphology is similar to that as shown in Fig. 6both for regions A and B. Microsegregation forming at the stage ofmushy zone solidification is shown in Fig. 8. It can be found thatthe simulation results lie between the Scheil equation and leverrule just as expected, and with properly choosing parameter,a=0.0257 for region A and a=0.0685 for region B, the Brody-Flemings model can agree well with the simulation results.Compared with region A, the increase of Fourier number forregion B means an enhanced back diffusion in this region. At thestage analyzed here the thermal conditions are similar for bothregions, so the enhanced back diffusion should be attributed totheir different microstructures. The coarsening of second dendritearms in region B leads to an increase of arm spacing. This effectequals to increasing the Fourier number just as pointed by Vollerand Beckermann [7]. Furthermore, well developed dendrite arms

0.780.445

0.450

0.455

0.460

0.465

0.470

0.475

0.480

c L (a

t%)

simulation results in region A

simulation results in region B

modified Scheil (Eq. 8)

modified Lever rule (Eq. 12)

modified Brody-Flemings (Eq. 11)

fS0.80 0.82 0.84 0.86 0.88 0.90 0.92

Fig. 8. Liquid concentration profiles as a function of the solid fraction for the

solidification of mushy zone with the solid diffusion coefficient DS=1�10�12 m2/s.

lead to a larger area of solid/liquid interface per unit volume. Theamount of solute transferred by solid diffusion is proportional tothe interface area. So the effect of back diffusion is pronounced inregion B. This conclusion is consistent with the work of Du andJacot [12].

Furthermore the solid diffusion coefficients DS is varied from1�10�12 to 1�10�10 m2/s. The simulation results are shown inFig. 9. It can be found that with increasing DS, which results in anenhanced back diffusion, both in region A (Fig. 9(a)) and region B(Fig. 9(b)) the liquid concentrations decrease when the same solidfraction is reached. With properly choosing parameters, a=0.0257,0.0831, 0.114, 0.243, 0.292 for DS increasing from 1�10�12 to1�10�10 m2/s in region A, and a=0.0685, 0.115, 0.143, 0.255, 0.306

0.80

0.445

0.44

0.45

0.46

0.47

0.48

DS = 5x10-11 m2/s

DS = 1x10-11 m2/s

DS = 5x10-12 m2/s

DS = 1x10-12 m2/s

DS = 1x10-10 m2/s

0.82 0.84 0.86 0.88 0.90 0.92 0.94

0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94

fS

c L (a

t%)

fS

Fig. 9. Liquid concentration profiles as a function of the solid fraction with

different solid diffusion coefficients for solidification: in region A ((a)) and in

region B ((b) and (c)), the symbols are obtained from simulation, the solid lines in

(a) and (b) are the best fitting to the simulation results based on Eq. (7) and the

solid lines in (c) are based on Eq. (7) using the fitting parameter obtained from (a).

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J. Li et al. / Journal of Crystal Growth 311 (2009) 1217–12221222

for DS increasing from 1�10�12 to 1�10�10 m2/s in region B, thepredictions of Brody-Flemings model fit all the simulation results.It can be seen that the Fourier number a for region B is lager thanthat for region A with all solid diffusion coefficients, but theincreasing extent decreases with increasing DS. For region B wehave also drawn the concentration profile based on the Brody-Flemings model using the Fourier number obtained from regionA (Fig. 9(c)). We can find that, when the solid diffusion coefficientis small, there is a large deviation between the theoreticalpredictions and simulation results, but, when the solid diffusioncoefficient is large, the theoretical predictions still agree well withthe simulation results. This finding indicates that the effect ofdendritic morphology on microsegregation is more pronouncedwith decreasing the solid diffusion coefficient DS, but is negligiblewith increasing DS. Kraft and Chang [30] have found that the effectof dendrite arm coarsening on microsegregation is moreprominent with decreasing solidification time. Both the decreaseof solid diffusion coefficient in our simulation and the decrease ofsolidification time in the work of Kraft and Chang [30] result in alessened back diffusion. So it is expected that the dendriticmorphology effect on microsegregation is important when theback diffusion is weak, whereas for the strong back diffusion thedendritic morphology influence may be neglected.

4. Summary

In summary, by performing 2D phase-field simulations ofdendritic array growth and quantitatively comparing withanalytical microsegregation models, the influence of microstruc-ture morphology on microsegregation is clear shown. It is foundthat even under no back diffusion condition the complex dendriticmorphology still affects microsegregation extensively. The reasonsaccounting for this influence include the dendrite arm coarseningand inhomogeneity of interdendritic liquid concentration causedby various interface curvatures. When there is a moderate extentof back diffusion, the effect of dendritic morphology on micro-segregation equals enhancing the back diffusion. This effect is

pronounced when the back diffusion of the system is weak,otherwise it can be neglected.

Acknowledgements

The authors wish to acknowledge the support of the NationalNatural Science Foundation of China (Grant no. 50401013) and theDoctorate Foundation of Northwestern Polytechnical University.

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