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Acta Materialia 60 (2012) 1478–1493

Phase-field study of competitive dendritic growth of converginggrains during directional solidification

Junjie Li, Zhijun Wang, Yaqin Wang, Jincheng Wang ⇑

State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China

Received 12 October 2011; accepted 19 November 2011Available online 28 January 2012

Abstract

The microstructure evolution of grains with different orientations during directional solidification is investigated by the phase-fieldmethod. For converging dendrites, in addition to the usually accepted overgrowth pattern wherein the favorably oriented dendrites blockthe unfavorably oriented ones, the opposite pattern of overgrowth observed in some recent experiments is also found in our simulations.The factors which may induce this unusual overgrowth are analyzed. It is found that in addition to the difference in tip undercooling, thesolute interaction of converging dendrites, which has been ignored in the classical theoretical model, also has a significant effect on thenature of the overgrowth at low pulling velocities. Solute interaction can retard the growth of dendrites at the grain boundary (GB) andinduce a lag of these dendrites relative to their immediate neighbors, which gives the unfavorably oriented dendrite the possibility toovergrow the favorably oriented one. However, this unusual overgrowth only occurs when the spacing between the favorably orientedGB dendrite and its immediate favorably oriented neighbor decreases to a certain level through lateral motion. These findings canbroaden our understanding of the overgrowth mechanism of converging dendrites.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Directional solidification; Dendritic growth; Phase-field models

1. Introduction

High-performance turbine blades in single-crystal formare routinely produced by directional solidification ofsuperalloys. The competitive growth between various ori-ented grains at the initial stage is a key factor to obtainthe desirable h001i texture. Usually, the favorably orientedgrains, whose preferred crystalline orientation (h001i formost superalloys) is parallel to the thermal gradient direc-tion, lead and block the unfavorably oriented grains, whosepreferred crystalline orientation is misaligned with the ther-mal gradient direction.

The generally accepted model for competitive graingrowth is that proposed by Walton and Chalmers [1] andschematically summarized by Rappaz and Gandin [2,3] as

1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2011.11.037

⇑ Corresponding author. Tel.: +86 29 88460650; fax: +86 29 88491484.E-mail address: jchwang@nwpu.edu.cn (J. Wang).

shown in Fig. 1. This model is based on the analysis ofthe difference in dendrite tip undercooling. In directionalsolidification the dendrite tip motion must keep up withthe liquidus isotherm at the steady state. Hence the tipvelocity of the unfavorably oriented dendrite (UO den-drite) is larger than that of the favorably oriented dendrite(FO dendrite). It is recognized that a large growth rate cor-responds to a large undercooling according to the classicaldendrite growth kinetics model [4]. Therefore the UO den-drite should lie behind the FO dendrite. For converginggrowth (left side in Fig. 1) this small lag will cause the den-drite in grain B to be blocked by the dendrite in grain A1 atthe grain boundary (GB). In this case the GB will be par-allel to the thermal gradient direction since grain A1 alsodoes not develop new dendrites at the GB. In the case ofdiverging growth (right side of Fig. 1), new dendrite armsdeveloping from grain A2 can lead to overgrowth of grainB and the GB is thus inclined towards the misorientedgrain.

rights reserved.

Δz Δz Δzθ

Gradient

V Vθ =V/cos θ

θ

Liquidus

A1 A2B

Fig. 1. Schematic representation of the Walton–Chalmers model illus-trating the mechanism of competitive gain growth [2,3].

J. Li et al. / Acta Materialia 60 (2012) 1478–1493 1479

Although these classical theory descriptions have beenconfirmed by some experiments [5,6] and simulations[7,8], some recent studies [9–12] have shown inconsistentresults. In experiments to investigate the directional solidi-fication of seeded bicrystals, Zhou et al. [9] found that forconverging growth the UO dendrites were also able toovergrow the FO dendrites, which led to the suppressionof the best-aligned grain by the misaligned one. The sameunusual overgrowth result between the misaligned grainand well-aligned stray grains was also observed by Souzaand coworkers [10,11]. In bicrystal samples with divergingdendrites, Wagner et al. [12] found that the relatively well-aligned grain cannot develop new dendrite arms at the GBso that the GB is parallel to the misaligned grain but notinclined towards the misaligned grain as illustrated inFig. 1.

All of these inconsistencies indicate that the classicalWalton–Chalmers model is not entirely satisfactory. In thismodel the overgrowth behavior is judged through an anal-ysis of the difference in tip undercooling of FO and UOdendrites. However, there are no quantitative descriptionsof the relationship between the undercooling differenceand other parameters such as pulling velocity and misori-entation angle. The assumption about dendrite tip kineticsused in this model, i.e. a large growth rate corresponds to alarge undercooling, is only strictly valid for free growth, inwhich undercooling is driven by external thermalizationbut growth rate is a free variable. In directional solidifica-tion the steady growth velocity is ascribed to the pullingvelocity but the interface temperature or undercooling isa free variable. Both directional solidification experiments[13] and numerical models [14,15] have shown that, beforeincreasing with increased pulling velocity, the dendrite tip

undercooling would first decrease in the relatively low-velocity regime. Hence the assessment of the tip positionin the Walton–Chalmers model may be not suitable forlow-growth-rate situations. Moreover, the existing analysisof dendrite tip kinetics is based on axisymmetric dendrites.There is no test as to whether or not this analysis is suitablefor dendrites with inclined asymmetry. The interaction ofthe solute with the converging dendrite tip—which mayhave a considerable influence upon the relative dendritetip position at low pulling velocities—is also neglected inthe classical model.

The mechanism for the unusual overgrowth observedexperimentally [9–12] cannot be determined conclusivelyfrom the experimental data. Due to the complexity ofinclined dendrite growth and the solute interaction betweenneighboring dendrites, it is also very difficult to treat com-petitive dendrite growth in an analytical way. The phase-field model is a suitable tool to investigate this issue.Recently, phase-field simulations have been extensively car-ried out in predicting microstructure evolution in direc-tional solidification [16–23], and these simulations havegreatly deepened our understanding on the relationshipbetween microstructures and controlling parameters. How-ever, most of these studies are concerned with the micro-structure evolution in a single well-oriented grain. Onlyin the work of Chen et al. [17] has competition betweentwo differently oriented silicon grains in undercooled meltsbeen simulated. Chen et al. found two distinct competitionmechanisms, with either interfacial or kinetic dominance,depending on the undercooling. However, this finding isclearly not suitable for the directional solidification ofsuperalloys, since the interface morphology and the solidi-fication conditions are totally different. In addition, in thework of Eiken [24], which extended the phase-field modelto magnesium alloys, competitive dendritic growth duringdirectional solidification was simulated as an example todemonstrate the model’s ability. However, it only foundthe overgrowth of UO dendrite by the FO dendrite. Asfar as we know no simulation studies concerning the unu-sual overgrowth exist.

In this paper, we used the phase-field approach to studythe competition of converging dendrites during directionalsolidification. Various factors that may influence the com-petition results, especially the effects of solute interaction,were analyzed. We try to illustrate the effects of soluteinteraction on the growth behavior of converging den-drites, test whether these effects can cause the unusual over-growth and show how these effects result in the unusualovergrowth, rather than quantitatively reproduce theexperimental process. These findings can give some insightinto the mechanism of the unusual overgrowth mannerobserved in recent experiments [9–11].

2. Simulation method

In order to distinguish grains with different crystallineorientations, we have chosen the multiphase-field model,

Table 1Properties used in the simulation.

Parameter Variable Value

Melting point of Ni Tm 1728 KLiquidus slope m 4.74 K wt.%–1

Partition coefficient k 0.5Entropy of fusion DS 2 � 105 J m–3 K–1

Interface energy r 0.21 J m–2

Anisotropy strength c4 0.02Initial concentration c0 6.5 wt.%Liquid diffusivity DL 3 � 10�9 m2 s–1

1480 J. Li et al. / Acta Materialia 60 (2012) 1478–1493

which was originally established by Steinbach et al. [25–27]and further developed by Eiken et al. [28] and Kim et al.[29–32], and rewritten it in a suitable form for the issuestudied here. The phase state with /i = 1(i – 1) denotesthe same solid phase but with different crystalline orienta-tions, whereas /1 = 1 denotes the liquid phase. The sumconstraint

Pj=1. . .n/j = 1 is maintained throughout. We

define a step function si = 1 if /i > 0 and si = 0 otherwise.Then the number of nonzero phase-field variables coexis-ting at a given point is:

N ¼Xn

i¼1

si ð1Þ

In the dilute solution limit the evolution equation of thephase field [27,29,31] is given by:

@/i

@t¼ � 2

N

Xn

j–i

sisjMijdfd/i� df

d/jþ Dgij

!ð2Þ

where

dfd/i¼

Pnj–1

e2SL2r2/jþwSL/j

� �for i¼ 1

e2SL2r2/1þwSL/1þ

Pnj–1;i

e2SS2r2/jþwSS/j

� �for i – 1

8>>><>>>:

ð3Þ

Dgij¼0 for i – 1 and j – 1

6/1ð1�/1ÞDSðT m�T �mcLÞ for i¼ 1 and j – 1

�6/1ð1�/1ÞDSðT m�T �mcLÞ for i – 1 and j¼ 1

8><>:

ð4ÞThe diffusion equation with anti-trapping current thatguarantees equal chemical potential between solid and li-quid is given as [18,30]:

@c@t¼ r/1DLrcL

þr eSLffiffiffiffiffiffiffiffiffiffi2wSL

p ðcL � cSÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/1ð1� /1Þ

p @/1

@tr/1

jr/1j

� �ð5Þ

The diffusion in the solid phase is neglected since it is verysmall and has little effect on the problem studied here.c = /1cL + (1 � /1)cS is a mixture concentration of solidand liquid. The parameters, e and w, with subscripts SL

or SS, are determined from the interface energy r andinterface width 2n of the solid–liquid interface or solid–so-lid interface:

eSL ¼4

p

ffiffiffiffiffiffiffiffiffiffiffiffiffinSLrSL

p; wSL ¼

2rSL

nSLð6Þ

eSS ¼4

p

ffiffiffiffiffiffiffiffiffiffiffiffiffinSSrSS

p; wSS ¼

2rSS

nSSð7Þ

By adjusting the ratio between rSL and rSS, the wettingproperty of the GB can be adjusted. The formation ofthe GB is not of concern here and we have set rSS = 2.5rSL

and nSL = nSS.

The phase-field mobility Mij has a definite relationshipwith the interface mobility. Mij can be also written asMSL if i = 1 or j = 1, or as MSS otherwise. For the limitof infinite physical interface mobility, the phase-fieldmobility of the solid–liquid interface MSL is determinedas [30,31]:

MSL ¼8rDL

ffiffiffiffiffiffiffiffiffiffi2wSL

p

pDSmceLð1� kÞe3

SL

ð8Þ

For the GB a simpler relationship exits between MSS andGB mobility [32]. As our focus is the dendrite tip growthbut not GB motion, we have simply set MSS = 0.1MSL.

In the case of a 4-fold anisotropic solid–liquid interfacialenergy, the term eSL should be extended to:

eSL ¼ e0ð1� 3c4Þ 1þ 4c4

1� 3c4

ð@/1=@x0Þ4 þ ð@/1=@y0Þ4

½ð@/1=@x0Þ2 þ ð@/1=@y0Þ2�2

" #

ð9ÞThe x0 and y0 axes are chosen to be parallel to the preferredcrystalline orientation which rotates an angle a0 from theoriginal reference coordinates x and y. The following rela-tions are satisfied:

@/1=@x0 ¼ cos a0ð@/1=@xÞ þ sin a0ð@/1=@yÞ ð10Þ@/1=@y 0 ¼ cos a0ð@/1=@yÞ � sin a0ð@/1=@xÞ ð11ÞIn the anisotropic case the phase-field Eq. (3) should berewritten as:

dfd/i¼

Pnj–1

r e2SL2r/j

� �þ @x r/1r/jeSL

@eSL@ð@/1=@xÞ

� �þ@y r/1r/jeSL

@eSL@ð@/1=@yÞ

� �þ wSL/j

8><>:

9>=>; for i ¼ 1

r e2SL2r/1

� �þ wSL/1 þ

Pnj–1;i

eSS2r2/j þ wSS/j

� �for i – 1

8>>>>>><>>>>>>:

ð12Þ

Since what we concerned here is the overgrowth mecha-nism but not a quantitative criterion for a specific superal-loy, a binary approximation to a nickel-based superalloywas used (see Table 1), where the partition coefficients k

and the slope of the liquidus m were assumed to be con-stant. In order to decrease the primary dendrite spacingso as to reduce the computational size, a relatively largethermal gradient G is specified, G = 60 K mm–1. With theseparameters the critical pulling velocity for the stability of

J. Li et al. / Acta Materialia 60 (2012) 1478–1493 1481

the planar interface are calculated to be 5.84 lm s–1 at theconstitutional undercooling limit and 1.75 � 105 lm s–1 atthe absolute stability limit. In our simulations the rangeof pulling velocity V is from 60 to 1000 lm s–1 so that den-drite structure is obtained. It was shown in Ref. [33] that aproper choice of the interface width of the phase field 2n isrelated to the smallest relevant length scale of the problem,which is the tip radius for dendrite growth. Therefore alarge value of 2n can be used at low pulling velocity sincethe tip radius increases with decreased pulling velocity.In our studies discretization is Dx = Dy = 0.5 lm forV < 300 lm s–1 and Dx = Dy = 0.25 lm for V > 300 lm s–1

and interface width 2n = 5Dx. This discretization is acompromise between calculation time and the indepen-dence of the discretization. It has no influence on the com-petition results because both the FO and UO dendrites arecalculated simultaneously within one system.

Simulations start from spherical seeds at the bottom ofthe calculation domain. The temperature was set acrossthe calculation domain linearly according to the imposedtemperature gradient and decreased with cooling rate_T ¼ VG. Both the growth of a single-orientation dendritearray and two converging dendrite arrays have been simu-lated. Stochastic noise is introduced into the phase-fieldequation in order to cause fluctuations at the interface,which results in the development of second dendrite arms.The left and right boundaries are treated periodically forthe single-orientation case and adiabatically for the two-orientation case. No-flux boundary conditions are appliedto the bottom and top boundaries. In order to keep thedendrite in the calculation box, a moving frame method[18,21] is used when the grid number between the dendritetip and top boundary is less than 300. In each moving stepthe bottom region with a height of 300 grids is shifted andcut off. The values of the phase field and the concentrationfield in the shifted region are frozen in and saved. The totalsolidification structure can be obtained by combining allthe “frozen-in” domains with the computational domain.This treatment can save simulation time and ensure thatthe concentration at the top boundaries is not affected bydendrite growth; the dynamics of the groove will, however,be neglected, but this does not affect the dynamics of thedendrite tip.

a Vg G

(a) (a

Vg G(b)

Fig. 2. Dendrite growth direction with various misorientation angles at V =h0 = 20�, h = 15.7�, (d) h0 = 30�, h = 23.5�.

3. Results and analysis

3.1. Growth direction of inclined dendrites

Before studying the overgrowth of differently orienteddendrites, we first test the reliability of phase-field simula-tion for inclined dendrite growth where the growth direc-tion is misaligned with the heat flow direction. Thegrowth behavior in our simulation is compared with obser-vations of the directional solidification of transparentorganic alloys [34–36].

The sequences of interface shape obtained in our simu-lations of four different misorientation angles h0 are shownin Fig. 2, where the pulling velocity V equals 60 lm s–1 andthe primary dendrite spacing k1 is 140 lm. It can be seenthat the actual growth direction of dendrites Vg liesbetween the thermal gradient direction G and the preferredcrystalline orientation a when h0 is not zero. This is consis-tent with experimental observations [34,35]. Moreover,changes of dendrite morphology with increased misorienta-tion angle observed in experiments [34,35], i.e. dendriteasymmetry and development of secondary branchesbetween dendrites, are also well reproduced in oursimulations.

It has been found that the growth direction will rotatefrom the thermal gradient direction to the preferred crys-talline orientation as the growth velocity or primary armspacing is increased [34–36]. These phenomena are alsoobserved in our simulations. In Fig. 3a1–a3 only the pull-ing velocity is increased. We can see that the growth direc-tion gets close to the preferred crystalline orientation as thevelocity is increased. A similar rotation of the growth direc-tion is found when increasing the primary spacing(Fig. 3b1–b3). Deschamps et al. [35] have reported thatdendrites turn their growth direction in order to satisfythe following relations:

h ¼ h0 �h0

1þ aPeb ð13Þ

where Pe = Vk1/D is the Peclet number, b is a fittingparameter, and a is related with h0 and can also be takenas a fitting parameter. The variation of h with Pe ath0 = 30� in our simulations is plotted in Fig. 4. We find thatthe simulation results can be fitted well by Eq. (13) with

aGVg

θ0

c)a

Vg G θ

(d)

60 lm s–1 and k1 = 140 lm: (a) h0 = 0�, h = 0�, (b) h0 = 10�, h = 7.7�, (c)

(a1) (a2) (a3)

(b1) (b2) (b3)

aVg G

θ=26.8°

aVg G

θ=22.1°

aVg

Gθ=28.8°

aVg G

θ=26.5°

aVg G

θ=24.1°

aVg G

θ=22.1°

Fig. 3. Evolution of growth directions with increasing pulling velocity at afixed primary spacing k1 = 120 lm (a1–a3) or with increasing primaryspacing at a fixed velocity V = 60 lm s–1 (b1–b3). The misorientationangle is fixed at h0 = 30�. (a1) V = 60 lm s–1, (a2) V = 120 lm s–1, (a3)V = 300 lm s–1, (b1) k1 = 120 lm, (b2) k1 = 160 lm, (b3) k1 = 220 lm.

0 2 4 6 8 10 12 1420

22

24

26

28

30

Phase field simulation

fitting to Eq. (13)

θ0=300

a=0.76

b=1.47

θ ( 0 )

Pe

Fig. 4. Variation of growth direction with Peclet number, Pe = Vk1D.

1482 J. Li et al. / Acta Materialia 60 (2012) 1478–1493

fitting parameters of a = 0.76 and b = 1.47. These valuesare comparable with experiments [35] and other simula-tions [36]. All of these comparisons indicate that thegrowth behavior of inclined dendrites can be accuratelyreproduced in our phase-field simulations.

3.2. Growth of a single-orientation dendrite array

In this section the dendrite growth of a single grain issimulated. Three factors that may affect the overgrowthbehavior of converging dendrites are analyzed, which ishelpful in illustrating the mechanism for the unusual over-growth results observed experimentally [9–11].

Fig. 5. Steady-state dendrite shape at V = 60 lm s–1 and k1 = 120 lm with dh0 = 30�.

3.2.1. Variations of tip undercooling

The tip undercooling which corresponds to the tip posi-tion in the thermal gradient is a key factor in determiningthe overgrowth manner, since the leading tip with smalltip undercooling usually blocks the lagging one with largetip undercooling. Hence the variation of tip undercoolingwith misorientation angle is first detected to see whetherthe UO dendrite array can be ahead of the FO dendrite.Steady-state dendrite shapes calculated at V = 60 lm s–1

and k1 = 120 lm for different misorientation angles h0 areshown in Fig. 5. It can be seen that the tip position lowersas h0 increases, and thus the tip undercooling raises withincreasing h0. The same results are also obtained at highervelocities (V = 300 lm s–1, 1000 lm s–1). However, oursimulations in the next section indicate that the overgrowthof FO dendrite by UO dendrite can happen only at lowvelocities (e.g. V = 60 lm s–1), but not at high velocities(e.g. V = 1000 lm s–1). Hence considering only the varia-tion of tip undercooling with misorientation cannot explainthe unusual overgrowth phenomenon.

3.2.2. The effect of dendrite spacing

Another factor that may influence the overgrowthbehavior is the primary dendrite spacing. It has been dem-onstrated both experimentally [37–39] and theoretically[40,41] that a wide range of dendrite spacing k1 can exist

ifferent misorientation angles: (a) h0 = 0�, (b) h0 = 10�, (c) h0 = 20�, (d)

(a)

(b)

10.50

10.75

11.00

11.25

11.50

11.75

12.00

S2

S4

S3

S1

λ1 μm

00

100

200

300

Δ T K

S4S1 S3 S2

30 35 40 45 50 55 60

100 120 140 160 180 200 220 2407.5

8.0

8.5

9.0

9.5

10.0

10.5

11.0

S4S3S2

00

100

200

300

ΔT K

λ1 μm

S1

S1 S2 S4 S3

Fig. 6. Variations of tip undercooling with dendrite spacing for different misorientation angles: (a) V = 60 lm s–1, (b) V = 1000 lm s–1. The steady-statemicrostructures for the spaces labeled S1–S4 are also given.

J. Li et al. / Acta Materialia 60 (2012) 1478–1493 1483

at fixed controlling parameters, V and G. The experimentalobservation [39] and simulations [14,18] also indicated thatthe tip undercooling decreases with increased k1. Hence thetip undercooling of a UO dendrite array with large spacing

may be smaller than that of FO dendrites with smallspacing. The variations of tip undercooling with dendritespacing for different misorientation angles are shown inFig. 6a and b, where V = 60 lm s–1 and V = 1000 lm s–1,

A B

(b)

A B

(a)

Fig. 7. Steady-state dendrite shape with nonuniform dendrite spacing in: (a) favorably oriented array, (b) unfavorably oriented array. Here the dendritegrowth velocity is equal to the pulling velocity, V = 60 lm s–1, but the lateral movement that will result in a uniform spacing still requires a long time.

0.076

0.078

0.080

0.082

c

distance μm

distance μm

0o 10o 20o 30o

0o 10o 20o 30o

(a)

0 100 200 300 400

0 20 40 60 80 100

0.064

0.068

0.072

0.076

0.080

c

(b)

Fig. 8. The lateral concentration distribution for differently orienteddendrite arrays at the same height as the tip position of a favorablyoriented dendrite array. The dots in these figures represent the points withmaximum solute gradient. (a) V = 60 lm s–1, k1 = 120 lm; (b)V = 1000 lm s–1, k1 = 30 lm.

1484 J. Li et al. / Acta Materialia 60 (2012) 1478–1493

respectively. Steady-state dendrite shapes with definedspacings are also given here. It can be seen that the decreaseof tip undercooling with spacing is more significant at lowvelocity. This is because that the solute diffusion length ld= D/V increases with the decrease in velocity. Hence in thelow-velocity regime the solute interaction is more intensiveand its influence on tip undercooling is more significant.Fig. 6a also indicates that when the spacing of the misa-ligned dendrite array is larger than that of the alignedone, its tip undercooling can be smaller and its tip positioncan be higher (see the points marked in Fig. 6a and the cor-responding steady-state dendrite structures). But this willnot happen at high velocities (Fig. 6b).

It should be noted that all the simulations in Fig. 6 arecarried out for a single-orientation dendrite array with uni-form dendrite spacing. The conclusion that the UO den-drite array with large spacing may be ahead of the FOone is obtained only by comparing the individual growthbehavior of differently oriented dendrites. However, thisconclusion may not be valid during the impingement oftwo converging dendrites. Fig. 7 provides some insight intothis problem. The simulation conditions in Fig. 7 are thesame as those in Fig. 6a except that the dendrite spacingis not uniform. The spacing between dendrite A and B islarger than the others. In this case the dendrite tip velocityhas been equal to the pulling velocity, but a long time is stillneeded to reach a uniform spacing through lateral motion.It can be seen that, for the FO dendrite array (Fig. 7a),both dendrites A and B are ahead of others, but for theUO dendrite array (Fig. 7b), only dendrite B is ahead. Thisindicates that the growth of UO dendrite is more sensitiveto the spacing variation in the side toward which it drifts.Therefore, during the impinging of two converging den-drites, even though the tip position of UO dendrite is aheadin the beginning, it may fall down when it gets close to theFO dendrite, since the spacing on this side, i.e. the sidetowards which the UO dendrite drifts, will decrease. There-fore we cannot predict the overgrowth results only by ana-lyzing the dendrite growth behavior of a single grain.Simulations of two converging dendrite arrays are neededto explore the manner of the overgrowth.

3.2.3. The effect of solute interactionIn addition to the factors discussed above, the effect of

solute interaction of converging dendrite tips has been

(a)

A B

A1A2A3 B1B2 B3

240 μm

B4

(b)

B1B2A1A2

t=12 s

100 μm (c)

A1A2 B2 B3

t=20 s

(e)

A1A2B2

B3

t=28 s(d)

A1A2 B2 B3

t=24 s

Fig. 9. Competitive growth of converging dendrites at V = 60 lm s–1 with uniform spacing, k1 = 120 lm: (a) the total structure by combining all the“frozen-in” domains with the calculated domain; (b–e) the concentration field near the tip region at different times.

J. Li et al. / Acta Materialia 60 (2012) 1478–1493 1485

assumed to be the most likely reason for the unusual over-growth results [9]. Although this effect also cannot be pro-ven only through the simulation of a single grain, someindirect explanations can be obtained in this way by com-paring the solute field of a differently oriented dendritearray. The lateral concentration distribution just ahead ofthe tip of the FO dendrite array is drawn forV = 60 lm s–1 (Fig. 8a) and V = 1000 lm s–1 (Fig. 8b).The lateral concentration distributions for UO dendritearrays with various h0 at the same height as the tip of theFO dendrite are also drawn here. The black dots in theconcentration curve of the FO dendrite array representthe points with maximum solute gradient. It can be seenthat when the pulling velocity is low (Fig. 8a), the lateralconcentration of the UO dendrite array is much closer tothat of the FO dendrite array, but the difference in lateralconcentration between FO and UO dendrite arrays is sig-nificant when the pulling velocity is high (Fig. 8b). InFig. 8a the maximum concentration for a dendrite arraywith h0 = 10� even exceeds the maximum solute gradientpoint in the concentration curve of the FO dendrite array.Therefore, it can be expected that at low pulling velocitiesthe concentration field of the two converging dendrites atthe GB will overlap significantly and the lateral solute dif-fusion between them be more difficult, so not only the UOone of the two converging dendrites, but also the FO onewill slow down and fall behind other dendrites. However,

this solute interaction effect should be weak at high pullingvelocities, especially for the FO one of the two convergingdendrite, since the solute diffusion length is short and thesolute field overlap is insignificant.

3.3. Growth of two converging dendrite arrays

In this section the dendritic structure evolution in a sys-tem containing two converging grains, FO dendrite andUO dendrite with misorientation angle of h0 = 10�, is sim-ulated. The competition of converging dendrites at differentconditions is analyzed so as to reveal the mechanism of theunusual overgrowth.

3.3.1. Competitive growth at low velocity with uniform

spacing

The competitive dendrite growth of two converginggrains (A and B) at V = 60 lm s–1 with uniform dendritespacing of k1 = 120 lm is shown in Fig. 9. The total solidi-fication structure (Fig. 9a) is obtained by combining the“frozen-in” domains with the computational domain. Itcan be seen that all the UO dendrites (B1, B2, B3 and B4)are blocked by the FO dendrite A1 at the GB. Meanwhiledendrite A1 does not develop new stems toward grain B.Hence, the GB remains parallel to dendrite A1. This resultis consistent with the prediction of the Walton–Chalmersmodel. In addition to the final structure, the overgrowth

A2 A1B4

B5

t=72 s

(e)

A2 A1 B4 B5

t=66.6 s

(d)

A2 A1 B4 B5

t=61.3 s

(c)(b)t=52 s

A2 A1B3 B4

100 μm

A B

A1

A2

A3B2

B1

B3

(a)

B4B5

240μm

Fig. 10. Competitive growth of converging dendrites at V = 60 lm s–1 when the dendrite spacing in grain A (k1 = 120 lm) is smaller than that in grain B(k1 = 140 lm). (a) The total structure by combining all the “frozen-in” domains with the calculated domain; (b–e) the concentration field near the tipregion at different times.

1486 J. Li et al. / Acta Materialia 60 (2012) 1478–1493

process is also given in Fig. 9b–e. The dendrite at the GB isreferred to as a “GB dendrite” in the following. At t = 12 sGB dendrite B1 has already largely fallen behind GB den-drite A1 and dendrite B2 begins to get close to GB dendriteA1. The tip position of dendrite B2 then lowers as the den-drite arm spacing between dendrite A1 and B2 decreases.This is consistent with the conclusion in Fig. 7, i.e. thegrowth of UO dendrite is more sensitive to the spacingvariation in the side towards which it drifts. Moreover,the solute field overlap between dendrite B2 and dendriteA1 will retard not only the growth of dendrite B1, but alsothe growth of dendrite A1. As shown in Fig. 9d, the tipposition of dendrite A1 is behind its neighbor A2. However,this lag cannot be maintained. Dendrite A1 will recover tothe same height as dendrite A2 when the growth of dendriteB2 almost stops (see Fig. 9e). The overgrowth process of thefollowing UO dendrites (dendrite B3 and B4) is similar tothat shown in Fig. 9b–e.

3.3.2. Competitive growth at low velocity with nonuniform

spacing

Experimental observations [42,43] have indicated thatthe dendrite spacing increases with increasing misorienta-tion. Hence the spacing of the UO dendrite array can be lar-ger than that of FO dendrite array. Here we detect theconverging dendritic growth behavior in such a case. InFig. 10 the initial dendrite spacing in the UO grain,k1 = 140 lm, is larger than that in the FO grain A,k1 = 120 lm. Fig. 6a shows that if the FO dendrite arrayand UO dendrite array of h0 = 10� grow separately withthese dendrite spacing conditions, the tip undercooling ofthe UO dendrite will be smaller than that of the FO den-drite. The same results are obtained before the UO dendritegets close to the FO dendrite as shown in Fig. 10b and e,where the tip position of dendrite B4 or B5 is ahead ofGB dendrite A1. However, the UO dendrite will becomelower and fall behind the FO dendrite when it becomes a

(d)t=24 s

A1A2 B2 B3

(c)t=20 s

A1A2 B2 B3

(e)t=28s

A1A2 B2 B3

(b)t=12 s

A2 A1 B1 B2

100 μmA B

A1A2A3B2B1 B3

(a) 240 μm

Fig. 11. Competitive growth of converging dendrites at V = 60 lm s–1 when the spacing, k1 between dendrite A1 and A2 is smaller than others. (a) Thetotal structure by combining all the “frozen-in” domains and the calculated domain; (b–e) the concentration field near the tip region at different times.

J. Li et al. / Acta Materialia 60 (2012) 1478–1493 1487

GB dendrite, see Fig. 10c and d. Hence the UO dendrites areall blocked by the FO one. This indicates that the initial den-drite spacing in the UO dendrite array does not affect theovergrowth manner. The reason for this is that when theUO dendrite grows into the GB region and gets close tothe FO dendrite, the spacing in the side toward which itdrifts will decrease no matter how large the initial spacing is.

The effect of initial dendrite arm spacing in the FO den-drite array is explored in Fig. 11, where the spacing (here-after termed SP1) between GB dendrite A1 and itsimmediate neighbor A2 is 100 lm and the spacing betweenother dendrites in grain A is 140 lm. Our simulations of asingle FO dendrite array have shown that 100 lm is still inthe allowable range of dendrite spacing at V = 60 lm s–1.In Fig. 10a, although dendrite B1 is blocked by dendriteA1, dendrite B2 nevertheless overgrows dendrite A1. Thisresult differs from the prediction of the Walton–Chalmersmodel but is consistent with some recent experimentalobservations [9–11]. The detailed overgrowth process isshown in Fig. 11b–e. It can be seen that the growth ofGB dendrite A1 is retarded as the unfavorably orientedGB dendrite B1 gets close to it. This solute interaction-induced lag is also found in the above simulations (Figs.9d and 10d). However, here the GB dendrite A1 cannotrecover to the same height as dendrite A2 after dendriteB1 is blocked. When dendrite B2 becomes the GB dendriteand grows toward dendrite A1, although the tip position of

dendrite B2 is also lowered, it is still ahead of dendrite A1.Hence, dendrite B2 can finally overgrow dendrite A1.

The result in Fig. 11 indicates that the unusual over-growth phenomenon is related to the dendrite spacing inthe FO grain. When spacing SP1 is small, the UO dendritecan overgrow the FO dendrite. In Figs. 9 and 10 spacingSP1 is large, and hence the unusual overgrowth phenome-non is not observed. However, careful observations of themicrostructure in Figs. 9 and 10 show that the final spacingSP1 at the end of the calculations becomes smaller thanthat in the beginning. We therefore assume that the unu-sual overgrowth may also happen in Figs. 9 and 10 witha much longer growth time. This assumption is confirmedby an extended simulation time of 2 weeks. The final micro-structure calculated under the same conditions as Fig. 9 isshown in Fig. 12. It is divided into three parts, (a), (b) and(c), from bottom to top. We can see that not only dendriteA1 but also dendrite A2 are blocked by the UO dendrites.The lateral displacements of FO GB dendrite and its imme-diate neighbor are drawn in Fig. 13. It can be seen that thelateral displacement of GB dendrite, dendrite A1 inFig. 13a and dendrite A2 in Fig. 13b, oscillates and gradu-ally increases with time. However, the lateral motion of theimmediate neighbor dendrite, dendrite A2 in Fig. 13a anddendrite A3 in Fig. 13b, is insignificant compared to theGB dendrite. Therefore spacing SP1 decreases continu-ously with time. Finally when this spacing is small enough,

(a) (b) (c)

240μm

A2 A3 B8

B9 B10

B11

A B

A1

A2 A3

B5 B6

B7

A B

B8

A2 A3 B8

B9 B10

B11 B8

B2

B3

B4

B5

B5 A3

A2 A1

B6 B7

B1

A B

Fig. 12. Microstructure of converging dendrites, the calculations areperformed using the same parameters as in Fig. 9 but continued for a longtime. The final structures are divided into three parts, (a), (b) and (c), frombottom to top.

0

5

10

15

20

25

30

t (s)

Δx (μ

m)

tip of dendrite A1

tip of dendrite A2

(a)

0 20 40 60 80 100 120

120 140 160 180 200

0

5

10

15

20

25

30(b)

t (s)

Δx (μ

m)

tip of dendrite A2

tip of dendrite A3

Fig. 13. The lateral displacement of dendrite tips as a function of time forthe three favorably oriented dendrites marked in Fig. 12.

1488 J. Li et al. / Acta Materialia 60 (2012) 1478–1493

the UO dendrite overgrows the FO dendrite. Similar resultsare also obtained for the extended simulation time usingthe conditions of Fig. 10. Hence it can be expected thatthe unusual overgrowth phenomenon will always happenno matter how large the initial spacing SP1 is. In addition,considering the conclusion obtained from the analysis ofFig. 10, we think that the unusual overgrowth phenomenonis not restricted to the initial dendrite spacing.

3.3.3. Competitive growth at high velocity

The competitive dendrite growth of two converginggrains (A and B) at V = 1000 lm s–1 is shown in Fig. 14.The simulations at low velocity have indicated that theunusual overgrowth will occur quickly if the initial spacingSP1 is small. Hence here the dendrite spacing between den-drite A1 and dendrite A2, k1 = 30 lm, is set smaller thanthat between other dendrites, k1 = 60 lm. It can be seenthat although the spacing of 30 lm is close to the lowerspacing limit, GB dendrite A1 is not blocked by the UOdendrite all the time. The detailed overgrowth processshown in Fig. 14b–e indicates that the growth of dendriteA1 is barely affected as the UO dendrite gets close to it.This is different from the observation at low velocity (Figs.9–11). The lateral displacement of dendrite A1 and A2 ismeasured and drawn in Fig. 15. We find that in the

horizontal direction GB dendrite A1 remains almost sta-tionary or moves slightly toward the right side, but den-drite A2 continuously moves toward the left side. Thespacing between dendrite A1 and A2 therefore increaseswith time. This spacing change tendency is also differentfrom that at low velocities. These differences in growthbehaviors reveal that the solute interaction between GBdendrites is insignificant when the pulling velocity is high.

3.3.4. Mechanism of unusual overgrowth phenomenon

When two converging dendrites get close to each other,their concentration fields will overlap. The above resultsindicate that the solute interaction has a significant effecton the growth behavior of GB dendrites at low pullingvelocities. Here this effect will be analyzed in detail.

Fig. 16 shows the horizontal liquid composition distri-butions just ahead of the FO GB dendrite (dendrite A1)when UO GB dendrite (dendrite B2 or B3) gets close todendrite A1. Fig. 16a is obtained from the simulationshown in Fig. 9, while Fig. 16b is obtained from the simu-lation shown in Fig. 14. The peak on the left side corre-sponds to the position ahead of dendrite A1, and thepeak on the right side corresponds to the position in front

100μm(a)

A B

A1A2

A3 B2B1 B3

B4

AB

(f)

t=0.875s

A1A2 B3 B4

(b) 25μm

t=0.958s

A1A2 B3 B4

(c)

t=1.042s

A1A2 B3 B4

(d)t=1.125s

A1A2B3

B4

(e)

A B

(g)

Fig. 14. Competitive growth of converging dendrites at V = 1000 lm s–1: (a) the total structure; (b–d) the concentration field near the tip region atdifferent times; (f) enlargement of grain boundary region; (g) experimental observation from Ref. [6].

0.0 0.5 1.0 1.5 2.0

-4

-2

0

2

4

6

tip of dendrite A1

tip of dendrite A2

Δ x (μ

m)

t (s)

Fig. 15. Variation of lateral displacement of dendrite tips with time forfavorably oriented GB dendrite A1 and its immediate neighbor A2marked in Fig. 14.

J. Li et al. / Acta Materialia 60 (2012) 1478–1493 1489

of dendrite B2 (or B3). It can be seen that in the case of lowvelocities (Fig. 16a), the liquid composition in front of den-drite A1 increases with time, while in the case of high veloc-ities (Fig. 16b), the liquid composition distribution aheadof dendrite A1 is almost unchanged except for a smallincrease in the tail on the right-hand side. This is due tothe larger diffusion length at low velocity than that at high

velocity. The severe superposition of the concentration fieldat low velocities hinders lateral solute diffusion. Hence thesolute concentration in the liquid around dendrite A1increases with time as shown in Fig. 16a. Since the drivingforce for dendrite growth decreases as the liquid concentra-tion increases, the growth of dendrite A1 slows down. Den-drite B2 goes through a similar process as dendrite A1, i.e.its tip lowers continuously as it gets close to A1. It shouldbe noted that the concentration curve in Fig. 16 is obtainedalong the line just ahead of dendrite A1 at different times,but not just ahead of dendrite B2 (or B3), so the right-handside peak of the concentration curve lowers with time.

The solute interaction effect can retard the growth of GBdendrites and induce a lag of GB dendrites relative to theirimmediate neighbors at low velocity. However, sometimesthe lag of the FO GB dendrite persists after the UO den-drite is blocked, but sometimes it cannot be maintained.In the following we will explain this difference by a carefulanalysis of the concentration distribution. The concentra-tion distributions for these two cases are shown inFig. 17, where a1–a3 correspond to Fig. 11b and b1–b3correspond to Fig. 10d. The isolines of liquid concentrationand the solute diffusion boundary for dendrite A1 areshown in Fig. 17a1 and b1. Here the solute diffusionboundary is defined as the places where the lateral concen-tration gradient is zero. This concept is inspired by the

100 150 200 250 3007.6

7.7

7.8

7.9

8.0

8.1

8.2

8.3(a)

Distance (μm)

c (w

t.%)

Tip A1

Tip B2

t=16 s

t=18 s

t=20 s

t=22 s

40 60 80 100

6.4

6.6

6.8

7.0

7.2

7.4

7.6

7.8

8.0

8.2(b) Tip A1

Tip B3

t=0.875 s t=0.958 s t=1.000 s t=1.042 s

Distance (μm)

c (w

t.%)

Fig. 16. Liquid composition distributions along the horizontal line justahead of the favorably oriented GB dendrite (dendrite A1) at differenttimes: (a) the results for the condition shown in Fig. 9; (b) the results forthe condition shown in Fig. 14.

1490 J. Li et al. / Acta Materialia 60 (2012) 1478–1493

work of Hunt and Lu [14,15] for analyzing the stabilitylimit of dendrite spacing. If the diffusion boundaries infront of the dendrite are converging, as per the two solidlines shown in Fig. 17a1, this dendrite will grow slowerthan its neighbors since the lateral diffusion for this den-drite is more difficult. If the diffusion boundaries in frontof the dendrite are diverging, as per the solid lines shownin Fig. 17b1, the growth of this dendrite will be faster thanthe growth of its neighbors. The concentration along thetwo dash-dot lines in Fig. 17a1 and b1 are shown inFig. 17a2 and b2, respectively. From these curves the con-centration gradient can be obtained as shown in Fig. 17a3and b3. Then the positions of solute diffusion boundary,m1, m2, n1 and n2, can be fixed. Following the sameapproach, the whole solute diffusion boundary can beobtained. In addition, this method, by connecting theinflection points of the concentration isoline, can alsoapproximately give the solute diffusion boundary. It canbe seen that in Fig. 17a1 the right and left solute diffusionboundaries get close to each other along the upward direc-tion, but they are separated in Fig. 17b1. This difference

may explain why the GB dendrite A1 can recover to thesame height as dendrite A2 under the condition corre-sponding to Fig. 10d, but is blocked under the conditioncorresponding to Fig. 11b.

When the FO GB dendrite cannot recover to follow thegrowth of its FO neighbor, it will be blocked by the nextapproaching UO dendrite, i.e. the unusual overgrowthoccurs. Our simulations indicate that this happens onlywhen the spacing SP1 is small. This explains why the soluteinteraction effect cannot induce the unusual overgrowthevery time two dendrites impinge. But how can the unusualovergrowth occur if the initial spacing SP1 is large? The lat-eral motion mechanism gives an answer to this question.This mechanism was first reported by Han and Trivedi[38] from observations of the directional solidification oftransparent organic material. They found that in a single-oriented dendrite array, the dendrite arm spacing willadjust to be uniform through lateral motion after the elim-ination or creation of dendrites. The reason for this lateralmotion lies in the asymmetry solute field on the two sides ofthe dendrite. Just as shown in Fig. 18a, the isoline of con-centration is flatter on the side of small spacing, which cor-responds to a small solute gradient. Hence solute diffusionon the small spacing side is more difficult than on the otherside. The dendrite will move toward the side with largerspacing. Similar process will happen when the UO dendritegets close to the FO dendrite, since the overlap of solutefields results in an asymmetric solute field on the two sidesof FO dendrite, just as shown in Fig. 18b. However, in thiscase the effect of the UO dendrite will stop when it isblocked, and the FO dendrite will move to the right sideto some extent until the next UO dendrite gets close again.Hence in Fig. 13 the lateral motion of GB dendrite oscil-lates with time, but spacing SP1 decreases continuouslyas the UO dendrites get close to the GB, one by one. There-fore, no matter how large the initial spacing SP1 is, theunusual overgrowth can occur finally.

To sum up, the solute interaction retards the growth ofGB dendrites and induces a lag of these dendrites relativeto their immediate neighbors. The solute interaction alsocauses the FO GB dendrite to move toward its FO neigh-bor. If spacing SP1 decreases to some extent through thislateral motion, the lag of the FO GB dendrite will persistafter one UO dendrite is blocked, and then it can beblocked by the next approaching UO dendrite, i.e. the unu-sual overgrowth occurs.

4. Discussions

The effect of solute interaction on the growth behaviorof GB dendrites, which has been ignored in the classicaltheoretical model, has been clearly shown by our phase-field simulations. These solute interaction-induced growthbehaviors are consistent with experimental observations[6,9,12]. The retarded growth of GB dendrites observedin our simulations at low velocities is also found indirectional solidification experiments by analyzing the

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

n2n1

m2

m1

line1 line2

Gc

(wt.%

/μm

)

(b3)

7.6

7.7

7.8

7.9

8.0

8.1

8.2

8.3

n2

n1

m2

m1

line1 line2

c (w

t.%)

(b2)

(a1) (b1)

(a2)

7.5

7.6

7.7

7.8

7.9

8.0

8.1

8.2

8.3

8.4

n2

n1

m2

line1 line2

distance (μm)

distance (μm)distance (μm)

distance (μm)

c (w

t.%)

m1

50 100 150 200 250 300

50 100 150 200 250 300100 120 140 160 180 200 220 240 260 280 300

100 120 140 160 180 200 220 240 260 280 300-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

n2

n1m2

m1

Gc

(wt.%

/μm

)

line1 line2

(a3)

Fig. 17. Concentration contours and the solute diffusion boundary (blue solid line), where the lateral concentration gradient is zero, for the favorablyoriented dendrite A1 being overgrown (a1) or surviving (b1), the variations of concentrations along line 1 and line 2 are given in (a2) and (b2)corresponding to (a1) and (b2), the relevant variations of lateral concentration gradient are given in (a3) and (b3). (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

J. Li et al. / Acta Materialia 60 (2012) 1478–1493 1491

microstructure of quenched samples [9]. For the occurrenceof unusual overgrowth the relative position of the dendritetips shown in Fig. 10 is also consistent with the experimen-tal observations in Ref. [9]. Furthermore, like the observa-tion in Ref. [9], our simulation also indicates that theunusual overgrowth only happens after several UO den-drites have been blocked, but not for every impingement.When the pulling velocity is high, although the growth ofthe FO GB dendrite is not slowed down, solute interaction

influences the development of secondary arms. The second-ary dendrites in both FO and UO dendrites have been sup-pressed at the GB, see Fig. 14. The microstructure obtainedfrom simulation (Fig. 14f) is comparable with thatobserved in experiments [6,12] (Fig. 14g). Hence the growthbehavior of converging dendrites in our simulations quali-tatively agrees with that in experiments.

Due to the 2-D nature and the simple treatment of mate-rial and controlling parameters, our current simulation

A1 B1

(b)

A1 A2

(a)

Fig. 18. Schematic illustration of the lateral motion mechanism for spacing adjustment: (a) single-orientation dendrite array; (b) converging dendrites.

1492 J. Li et al. / Acta Materialia 60 (2012) 1478–1493

cannot quantitatively reproduce the experimental process.However, all the qualitative agreements between the simu-lation and the experiment indicate that the solute field evo-lution in our simulation is a proper 2-D simplification forreal experiments. The effect of solute interaction on theGB dendrite growth behavior, which results in the unusualovergrowth in our 2-D simulation, should also exist in 3-D,as there is no essential different in 2-D or 3-D for the exis-tence of solute interaction. Although the degree of soluteinteraction in 2-D is different from that in 3-D, this doesnot mean that the 2-D simulation is useless in revealingthe mechanism of microstructure evolution. For the solutediffusion-controlled process, the basic mechanism revealedin 2-D simulation normally also exists in 3-D, which hasbeen found in many simulation works, e.g. in the studyof dendrite spacing [18], sidebranching [22] and twinneddendrite tips [44]. In these studies, although the quantita-tive results are different between 2-D and 3-D, the basicmechanism and rules are the same. In experiments, the sol-ute field overlap will also definitely happen when two con-verging dendrites get close to each other. Hence the basicmechanism revealed in our simulation should also exist inexperiments, and at least the explanation for the unusualovergrowth obtained from our simulation is reasonable,even if at present we cannot fully confirm that this is thespecific reason for the experimental observation. Simula-tions in 3-D should be carried out to acquire a betterunderstanding of unusual overgrowth. More experimentsare also needed to determine the critical pulling velocityand thermal gradient for the occurrence of the unusualovergrowth, and to test whether such phenomenon canhappen in 2-D. With progress both in simulation andexperiment, a thorough understanding of this phenomenoncan be achieved.

Latent heat is released during solidification. Just likeconcentration field overlap, there should be also heat flowsuperposition at the GB region for the converging den-drites. Such heat flow interaction will induce curved ther-mal fields and influence the dendritic growth behavior.Hence it may be also a factor influencing the overgrowth

manner. However, D’Souza et al. [6] have shown that thelatent heat contribution is only �10% of the total heat fluxusing typical processing conditions (G � 5 K mm�1,V � 10�5 m s�1) and physical constants for Ni-base super-alloys. Therefore, it could be expected that the effect oflatent heat release on the heat flow is insignificant. In fact,in experiments where unusual overgrowth is observed [9],the quenched sample obtained by the liquid metal coolingtechnique shows a near-horizontal interface with maximumslope at the mold wall of 4�. This indicates that the influ-ence of heat flow interaction in the thermal field is nearlynegligible, especially near the small region where the con-verging dendrites impinge. It is thus a reasonable approxi-mation to consider the solute interaction effect in theabsence of any heat flow interaction. Furthermore, sincethe thermal diffusion length is several orders of magnitudelarger than solute diffusion length, in the small length scaleof several hundred micrometers, the effect of solute interac-tion on dendritic growth should be more important thanthat of heat flow interaction. It can be expected that thesolute interaction is the main reason for unusualovergrowth.

5. Conclusions

The inclined dendrite growth in a single grain and thecompetitive dendrite growth of two converging grains aresuccessfully simulated by the phase-field method. The effectof solute interaction on the growth behavior of GB den-drites, which has been ignored in the classical theoreticalmodel, has been clearly demonstrated in our simulations.This effect can result in retarded growth of dendrites atGBs, suppression of second dendrite arms and lateralmotion of FO GB dendrites. It has been shown that atlow pulling velocities the lag of FO GB dendrite can persistafter one UO GB dendrite is blocked, and induces over-growth by the next UO dendrite. However, this unusualovergrowth only happens when the spacing between theFO GB dendrite and its immediate FO neighbor decreasesto a certain level via lateral motion. These findings provide

J. Li et al. / Acta Materialia 60 (2012) 1478–1493 1493

an explanation for recent experimental observations [9], i.e.at low pulling velocities the UO dendrite can overgrow theFO dendrite, though this only happens after several UOdendrites have been blocked, but not for every impinge-ment. Although our simulation is based on 2-D and simpleparameters are used, considering the solute field overlap isa universal process during dendrite impingement both insimulation and in the experiments, our explanations forthe unusual overgrowth based on the analyses of soluteinteraction is reasonable and is useful in helping to eluci-date various aspects of this phenomenon. Simulations in3-D space are needed for further exploration of this area.

Acknowledgements

The work was supported by National Natural Sciencefoundation of China (Grants Nos. 51071128, 51101124),National Basic Research Program of China (Grants No.2011CB610401), Program for New Century Excellent Tal-ents in University, the Fundamental Research Fund ofNorthwestern Polytechnical University (JC201006), FreeResearch Fund of State Key Laboratory of SolidificationProcessing (67-QP-2011). We also thank Dr. Zhou Yizhoufor the valuable suggestions.

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