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Acta Materialia 60 (2012) 5501–5507

Orientation selection process during the early stage of cubicdendrite growth: A phase-field crystal study

Sai Tang a, Zhijun Wang a, Yaolin Guo a, Jincheng Wang a,⇑, Yanmei Yu b, Yaohe Zhou a

a State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Youyi Western Road 127, 710072 Xi’an, Chinab Institute of Physics, Chinese Academy of Science, PO Box 603, 100190 Beijing, China

Received 17 May 2012; received in revised form 3 July 2012; accepted 3 July 2012Available online 13 August 2012

Abstract

Using the phase-field crystal model, we investigate the orientation selection of the cubic dendrite growth at the atomic scale. Our sim-ulation results reproduce how a face-centered cubic (fcc) octahedral nucleus and a body-centered cubic (bcc) truncated-rhombic dodeca-hedral nucleus choose the preferred growth direction and then evolve into the dendrite pattern. The interface energy anisotropy inherentin the fcc crystal structure leads to the fastest growth velocity in the h100i directions. New {111} atomic layers prefer to nucleate atpositions near the tips of the fcc octahedron, which leads to the directed growth of the fcc dendrite tips in the h100i directions. A similarorientation selection process is also found during the early stage of bcc dendrite growth. The orientation selection regime obtained byphase-field crystal simulation is helpful for understanding the orientation selection processes of real dendrite growth.� 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Orientation; Dendrite formation; Crystal growth; Computer simulation; Phase-field crystal model

1. Introduction

Orientation selection of the dendrite tip – a crucial issuein the field of dendrite growth – not only produces variousdendrite patterns like fourfold and sixfold dendrites, butalso greatly determines the performance of industrial mate-rials such as turbine blades and semiconductor materials[1,2]. In a two-dimensional (2-D) simulation of the den-dritic growth, square crystalline lattices and triangularcrystalline lattices present a fourfold symmetrical dendriteand a sixfold symmetrical dendrite, respectively, both withtips pointing in the direction corresponding to the crystal-line lattice’s maximum interface energy [3,4]. Generallyspeaking, body-centered cubic (bcc) and face-centeredcubic (fcc) dendrites grow along the h1 00i directions, whilehexagonal close-packed (hcp) dendrites grow along theh110i directions [1,5–8]. For the last few decades, scientistshave been trying to reveal the relationship between pre-

1359-6454/$36.00 � 2012 Acta Materialia Inc. Published by Elsevier Ltd. All

http://dx.doi.org/10.1016/j.actamat.2012.07.012

⇑ Corresponding author. Tel.: +86 29 8846 0650; fax: +86 29 8849 2374.E-mail address: jchwang@nwpu.edu.cn (J. Wang).

ferred growth direction and crystal structure, and, further,to demonstrate how a dendrite tip selects its preferredgrowth direction.

Normally, interface energy anisotropy and interfacekinetic anisotropy, two factors associated with crystalstructure, are used to elucidate the relation between pre-ferred growth direction and crystal structure. As early asthe mid-1980s, Ben-Jacob et al. [9,10] constructed a modelto elucidate the influences of the two factors on dendriteorientation selection. Specifically, the interface energyanisotropy dominates dendrite orientation selection whenthe growth driving forces are low, whereas the interfacekinetic anisotropy becomes dominant in determining thedendrite growth direction when the growth driving forcesare large [11,12]. In recent years, phase field simulationshave further demonstrated how the interface energy anisot-ropy and the interface kinetic anisotropy subtly influencedendrite growth directions [5,13,14]. For example,Haxhimali et al. [5] obtained continuous changing pre-ferred growth directions with composition-dependent inter-face energy anisotropy parameters. Bragard et al. [13]

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5502 S. Tang et al. / Acta Materialia 60 (2012) 5501–5507

studied the competition between the interface energyanisotropy and the interface kinetic anisotropy and its con-sequence on the dendrite patterns quantitatively. Amo-orezaei et al. [14] investigated the dendrite orientationselection controlled by the interplay between the interfaceenergy anisotropy, the thermal gradient, and the pullingvelocity in directional solidification. Previous studies haveshown that the preferred growth direction seems to be clo-sely related to the interface energy anisotropy and the inter-face kinetic anisotropy. However, a clear understanding ofthe atomic process by which a nucleus selects the preferredgrowth direction is still lacking. The final dendrite mor-phology, whether fourfold or sixfold, is attributed to theorientation selection process during the early stage of den-drite growth. A thorough description of the orientationselection process can not only provide a direct and visualimage of dendrite pattern formation, but may also enablethe optimization of processing conditions for controllingthe final morphology of the crystal and improve materialproperties. Nevertheless, related research on the orienta-tion selection process is very rare. Since the orientationselection process often occurs at the early stage of dendritegrowth, the nucleus’s shape before evolving into a dendriteand the formation of a dendrite tip are the key points inunderstanding the orientation selection process. The sur-face/volume ratio for the nanoscale nucleus is very high,and therefore the shape of the nucleus is greatly influencedby the interface energy anisotropy, which favors a polyhe-dron nucleus bounded by closest packed planes, ratherthan a sphere nucleus. For example, the fcc nanocrystallineis usually of octahedral shape [15–21], and can develop intoa dendrite with the dendritic tip pointing to the h100idirections [20,21]. The evolution from an octahedron intoa dendrite pattern at the early stage of dendrite growthinvolves an orientation selection process of the dendritetip. However, the atomic transformation process is notyet clearly understood.

Investigation of crystal growth at the atomic level canprovide the most direct and essential description of orienta-tion selection for dendrite growth. Nevertheless, it is stillnot possible to directly observe the atomic process experi-mentally. Numerical simulation, as an important currentresearch methodology, has greatly deepened the under-standing of dendrite growth. For instance, the Monte Car-lo method and the lattice gas model have been used tosimulate the dendrite growth of simple crystal structures,such as square and hexagonal lattices in two dimensionsand a simple cubic lattice in three dimensions [3,4,22,23].For other complex structures including bcc, fcc and hcpstructures, research on dendrite growth at the atomic scaleis very limited. Schulze [24] simulated three-dimensional (3-D) dendrite growth of an fcc structure in supercooled meltsby using the kinetic Monte Carlo method (KMC) at theatomic scale. The authors consider the dendrite growth tobe determined by the anisotropic surface diffusion on thecrystal surface. During the real dendrite growth, a long dis-tance diffusion process of mass from liquid phase to crystal

phase is generally supposed to play an important role [25].However, such long-range diffusion is not easily accessiblein the KMC simulation, considering the limited realisticcomputation time. As discussed above, the dendrite orien-tation selection process of fcc and bcc structures has notbeen revealed by experiment and numerical simulationyet. Thus, new research methods and models are expectedto shed some light on that problem.The phase-field crystal(PFC) model is derived from the classical density func-tional theory of freezing [26], and is capable of simulatingthe atomic growth process at the diffusion time scale.Anisotropy and elasticity [27–30] are also introduced intothe PFC simulations inherently [31–33]. In this work, bychecking the atomic growth process near the growing den-drite tip, we adopt the PFC model to investigate the rela-tionship between the crystal structure and the preferredgrowth direction of the dendrite tip, with emphasis onthe fcc dendrite growth process. The similar orientationselection regime of the dendrite tip will be examined duringthe early stage of bcc dendrite growth.

2. The phase-field crystal model

The PFC model is derived from the free energy func-tional [27,28]:

F ¼Z

d~xw2½eþ ð1þr2Þ2�wþ w4

4

� �; ð1Þ

and thus the dynamic equation is given by

@w@t¼ r2ðdF =dwÞ ¼ r2f½eþ ð1þr2Þ2�wþ w3g; ð2Þ

where w is a conserved order-parameter field related to thenumber density field q through w / ðq� qref

L Þ=qrefL , and

qrefL is the number density of the reference liquid; e is a phe-

nomenological parameter related to the temperature andbulk modulus of the liquid and crystal. The simulationparameters ð�w; eÞ are chosen based on the 3-D phase dia-gram [34], which contains fcc, bcc and hcp structures,where �w is the average value of w. The initial condition isthat a sphere nucleus, with a radius length of about six lat-tice constants, locates at the center of the simulation boxand is surrounded by homogenous liquid. The density fieldof the nucleus is given by a single-mode approximation:wfcc ¼ 8A½cosðqxÞ cosðqyÞ cosðqzÞ� þ �w for fcc structuresand wbcc ¼ 4A½cosðqxÞ cosðqyÞ þ cosðqyÞ cosðqzÞþ cosðqzÞ cosðqxÞ� þ�w for bcc structures, where A and q

are the amplitude and the wave number of the density fluc-tuation around an average density �w respectively. The ini-tial density of the liquid phase is �w. Eq. (2) is solved usingthe semi-implicit Fourier spectral method with grid spacedx = 1, and time step dt = 0.75 for fcc, and dt = 1.5 forbcc. All simulations are performed in a cubic box with sizeof L = 800. The simulation parameter �w is chosen in therange from �0.48 to �0.4725 with e = �0.53 for the fccstructure, and from �0.35 to �0.365 with e = �0.3 forthe bcc structure based on the one-mode phase diagram

Fig. 1. The layer-by-layer growth process of fcc crystal as simulated withðe; �wÞ ¼ ð�0:53;�0:4725Þ at different times: (a) t = 1628, (b) t = 1642, (c)t = 1703, and (d) t = 1778; their corresponding (010) cross-section imagesare shown in (e-h).

Fig. 2. The density maps of the fcc {111} faces at the nucleation stages(t1–t2) and the subsequent aggregation stages (t3–t4), where t1 = 1613,t2 = 1642, t3 = 1815, t4 = 1823, t5 = 1837, and t6 = 1852.

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[34]. Atomeye software [35] is employed to visualize theatom configuration.

3. Results and discussion

In this section, we will show that how the crystal shapeevolves from an fcc octahedron nanonucleus into a den-drite. The orientation selection of a bcc crystal will alsobe introduced briefly. We will then relate our simulationresults to experiments.

Fig. 1a–d shows the layer-by-layer growth process of anfcc crystal, as simulated with parametersðe; �wÞ ¼ ð�0:53;�0:4725Þ. As Fig. 1a shows, due to the factthat the growth velocity is fastest along the h10 0i direction,the initial sphere-shaped nucleus evolves into an octahe-dron bounded by eight {11 1} planes with six tips pointingto the h10 0i directions, where the connection of two {111}planes is along the h110i directions. As shown in Fig. 1b,the octahedron-shaped crystal continues to grow. Startingfrom the nucleation at an area near the octahedron tips,the newly absorbed atoms of four {111} faces around eachoctahedron tip construct one small {100} face, making theshape of the crystal evolve into a truncated octahedron.Here, new solid atoms in the newly nucleated {111} layersat areas near the octahedron crystal tips originate from thepreferred nucleation and directed growth modes, i.e. newatoms grow up on the existing solid, not the stochasticnucleation in the melt. While the nucleated {111} layersgrow up rapidly, new solid atoms are absorbed into thetruncated tip simultaneously, leading to the nucleation ofnew {100} layers on the existing {100} faces. Then, asshown in Fig. 1c and d, a sharp tip forms quickly andmoves forward, which results in the displacement of the{10 0} layers in the h100i directions, where, however, the{11 1} faces do not advance until nucleation of the next{11 1} layers. As shown in Fig. 1d, the large growth veloc-ity in the h100i directions causes the rapid re-formation ofthe sharp tip. The difference in the growth velocity of theh100i and h111i faces is due to the intrinsic anisotropyof the liquid–crystal interface energy of the fcc PFC struc-ture. In the planar growth of the {100} and {11 1} planesfor the fcc structure in PFC simulation with parameters ofðe; �wÞ ¼ ð�0:53;�0:4725Þ [33], the liquid–crystal interfaceenergy density c100 = 0.0113 for {100}, which is muchhigher than c111 = 0.0082 for the {111} plane. The fccstructure with parameter e = �0.53 in our simulation hasa large liquid–crystal interface energy anisotropye4 ¼ r100�r111

r100þr111¼ 0:159. In this case, the facet crystal with

octahedral shapes is expected [36]. Moreover, the liquid–crystal interface energy anisotropy contributes to the pre-ferred growth in the h100i directions to be of fcc crystal[37], which is necessary for the formation of fcc dendritewith tips pointing to the h1 00i directions.As mentionedabove, the nucleation of a new {111} layer near the tipson the {11 1} faces of the octahedron crystal directly givesrise to the small {100} faces forming at truncated-octahe-dron tips and triggers the growth of the new layer on the

{111} faces. Next, by checking the nucleation of the{111} layer, we will present the orientation selection pro-cess during the evolution from an octahedron fcc crystalto a dendrite with tips pointing to the h100i directions.Fig. 2 shows the density map of a new layer of the {11 1}face as shown in Fig. 1a: t1 and t2 are the moment beforeand after position A is occupied by atoms, respectively,and similarly, t3 and t4 for position B, t5 and t6 for positionO. Position A is firstly occupied by atoms, then position Band finally position O. As shown in Figs. 1b and 2, posi-tions near the tip of the octahedron crystal in the {111}planes are the preferred regions for a new layer nucleation,which is further confirmed by the growth of an octahedron-shaped crystal interfered by white noise in additional sim-ulations. In the simulation with white noise, the crystalgrowth still demonstrates a preferred nucleation and direc-ted growth mode, where the added random noise in themelt does not take significant effects. This indicates thatthe preferred nucleation and directed growth on the exist-ing solid prevail over the potential random nucleationinduced by the white noise in the melt. Therefore, the white

Fig. 4. The smoothed density profile of the fcc {111} along the dashedline AOB (as denoted in Fig. 2) during (a) nucleation, and (b) thesubsequent aggregation stages.

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noise seemingly does not play a role in the crystal growthof our simulations, i.e. does not change the preferred nucle-ation and the directed growth mode of the new crystal lay-ers. In the additional simulations, we use the PFC model byadding a white noise term g in Eq. (2) [27,28]:

@w@t¼ r2ðdF =dwÞ ¼ r2f½eþ ð1þr2Þ2�wþ w3g þ g; ð3Þ

where g is a stochastic and conserved noise, with zero meanand correlations: hgð~r; tÞgð~r0; t0Þi ¼ �Gr2dð~r �~r0Þdðt � t0Þ,and G denotes the strength of noise. Moreover, it has beenreported in previous theoretical and experimental research[38–40] that nucleation can also occur at the tips of thepolygon facet crystal in 2-D growth. In order to give a rea-son for the preferred nucleation of a new {111} layer al-ways at the corner of the existing {111} planes, we checkthe density and free energy density along line AOB (posi-tions A, O and B are the corner, the center and the middleof edge of the {111} plane, respectively). Fig. 3a shows thedensity profile of a new layer along line AOB when thenucleation of this layer just happens, as shown in the den-sity map at t2 in Fig. 2. In Fig. 3b, t0 = 1050 is the timewhen a nucleation is completed and t2 = 1642 is the timewhen the next nucleation happens. As Fig. 3b shows, thefree energy at position A is much lower than that at otherpositions on line AOB, demonstrating that the drivingforce for the solid atoms growing at position A is muchhigher than those at other positions along line AOB. Thedecrease in free energy at position A increases the probabil-ity of atoms overcoming the energy barrier which hindersatoms in the liquid phase from growing upwards on asmooth {111} plane. When the energy barrier is overcome,atoms grow upwards at position A very quickly, resultingin a sharp increase in density at position A as shown inFig. 5a from t1 to t2. Meanwhile, the free energy at positionO and B is nearly the same as that of the liquid phase, pre-venting a new atom from attaching into position O and Bbecause of a low driving force. In addition, from the per-spective of geometry, the nucleation position at tip A ofthe octahedron fcc crystal is favored by the final dendrite

Fig. 3. (a) The profile of the mass density (solid line) and its smoothedvalues (dotted line) of the fcc {111} faces along a line AOB (as denoted inFig. 2) at t2 = 1642, where the density smoothening is conducted using afast Fourier transform smoothing procedure (15 data points, a low-passcutoff frequency of 0.0333), and (b) the profile of the free energy densityalong the line AOB before nucleation of a new fcc (111) faces (t = 1050–1612) and at the nucleation stage (t = 1642).

pattern with tips pointing to the h100i directions. Never-theless, if the center O is the nucleation-favored position,a dendrite pattern with tips pointing to the h1 11i direction,instead of the common h100i type dendrite in naturalworld, will form.We have also checked the density diffusionduring the nucleation and growth process of a new layer.As Fig. 4a shows, during the nucleation process (from

Fig. 5. (a) The variation of the density at different positions along the lineAOB (as denoted in Fig. 2) vs. time, t1–t2, t3–t4, and t5–t6 correspond tothe positions A, O, and B respectively, and (b) the evolution of densitycorresponding to the successive layer nucleation processes.

Fig. 6. (a) The displacement of crystal tip during layer-by-layer growthwith ðe; �wÞ ¼ ð�0:53;�0:4725Þ, multi-layer growth withðe; �wÞ ¼ ð�0:53;�0:4675Þ, and dendrite growth withðe; �wÞ ¼ ð�0:53;�0:4607Þ. In the curve of displacement vs. time of layer-by-layer growth, the time from tL1 to tL2 is the incubation time (tN) of anucleation, while from tL2 to tL3 is the growth time (tG) cost by the layergrowth after nucleation, and (b) the corresponding crystal shape ofdendrite growth (A), multi-layer growth (B) and layer-by-layer growth(C), respectively.

S. Tang et al. / Acta Materialia 60 (2012) 5501–5507 5505

t = 1050 to t = 1612), the density along line AOB of a newlayer increases slowly, because the global density diffusionin the liquid phase has to undergo a large distance to theliquid–crystal interface. After this, a new cluster nucleatesat position A, which provides energetically favorable steppositions for new solid atoms to grow upwards, as shownin Figs. 1 and 2. However, the new solid atoms grow atthese steps so rapidly that the global density diffusion isno longer capable of providing enough atoms simulta-neously. Consequently, the local density diffusion fromthe liquid phase and growing layer around the growth siteplays a dominant role during the growth of a new layer.Being confined by the difficulties of demonstrating a 3-Ddiffusion field for the local density diffusion, we have tosimplify it by displaying the density profile evolution alongline AOB during the nucleation and growth process of anew layer. As denoted by the arrows in Fig. 4b, fromt = 1688 to 1740, with the advance of the steps into the cen-ter of the {111} plane (as Fig. 2 shows) along line AOB,the sharp increase in the density at the growth site (as theleft arrow denotes) is at the expense of the density of theregion near the place (as the right arrow denotes). Becausethe diffusion field is three-dimensional, the integral of thedensity profile along a single line such as line AOB inFig. 4b from t = 1688 to t = 1740 is not conserved, yetthe integral of density in the 3-D region is conserved. Thenew layer is fully filled soon after nucleation. In addition,we find that the nucleation of the {111} layers is deter-mined by the density diffusion from the liquid phase tothe liquid–crystal interface. The density diffusion processresults in an inhomogeneous distributed supersaturationaround the crystal surface such that the supersaturationat the corner (position A) of the {111} plane of the octa-hedral crystal is higher than that at the other positions inthe plane. The supersaturation in the PFC model is de-noted by r ¼ �w��wL

�wS��wL, where �wS and �wL are the solidus and liq-

uidus in the coexistence region of the phase diagramrespectively [34]. Our results are consistent with previousresults [38,39,41] in which there is a similar anisotropic dis-tribution of supersaturation from the apex to the center ofthe edge of a facet polygon when it develops into a 2-Ddendrite.Just as the preferred nucleation position at theapices of a polygon can lead to an instability of the polygonwith increasing supersaturation and then produce a den-drite pattern [38–40], the preferred nucleation positionsaround the tips of fcc octahedron crystal can also under-mine the stability of the octahedron crystal, and furthercause a typical fcc dendrite with increasing supersatura-tion. Fig. 5a shows the density evolution of a nucleationprocess at positions A, O and B along line AOB. InFig. 5a, the period from t0 to t2 is the incubation time tN

(t2 � t0 = 592) expended by the nucleation process, andthe period from t2 to t6 is the growth time tG

(t6 � t2 = 210) for filling the new {111} layer after nucle-ation while the time for nucleation is much longer. Thissuggests that the crystal growth velocity is limited by thenucleation of a new layer in the layer-by-layer growth

mode. Fig. 5b shows the periodic oscillation of densityvs. time at position A during successive nucleation pro-cesses in the layer-by-layer growth mechanism. Fromt = 0 to about t = 4500, the crystal shape remains an octa-hedron with supersaturation defined by simulation param-eters ðe; �wÞ ¼ ð�0:53;�0:4725Þ under the layer-by-layergrowth mechanism. Moreover, the growth mechanismand crystal shape greatly depend on supersaturation. AsFig. 6 shows, if the driving force increases by settingparameters in the range from ðe; �wÞ ¼ ð�0:53;�0:4725Þ to(�0.53, �0.4607), the incubation time (tN) for nucleationdecreases correspondingly to be shorter than the growthtime (tG), causing the next new {111} layer to nucleate be-fore the former layer is fully filled. Then the layer-by-layergrowth mode is replaced by the multi-layer growth mode,which facilitates the crystal to grow faster along theh100i directions than along the other directions, thenundermines the stability of the octahedron crystal, and fi-nally develops into an fcc dendrite with branches alongthe h100i directions [31,33].

A similar orientation selection regime is also obtained inthe bcc dendrite orientation process, as shown in Fig. 7.The precursor of the dendrite pattern of bcc crystal is trun-cated rhombic dodecahedron, consisting of 12 {11 0}planes with six tips pointing to the h10 0i directions andeight tips pointing to the h11 1i directions. The truncatedrhombic dodecahedron is attributed to the interface energyanisotropy of the bcc structure [19,42]. Although the nucle-

Fig. 7. The evolution of the shapes of the bcc PFC dendrite as simulatedwith ðe; �wÞ ¼ ð�0:3;�0:3625Þ: (a) the truncated-rhombic dodecahedronshape at 520 time steps, (b) the nucleation of new {110} layers at area nearthe center of {110} faces at 550 time steps, (c) nucleation on area of {110}planes near h100i tips at 5950 time steps during layer-by-layer growth,and (d) the layer-by-layer growth mode changes to the multi-layer growthmode at 10,400 time steps.

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ation position of a new layer is at the center of the {110}plane at the beginning, it will transit into the region of the{11 0} plane near the h100i tips with the increasing size ofthe growing crystal. Analogous to the evolution of the fcccrystal from octahedron into dendrite, the nucleation of anew {110} layer near the h1 00i tips on the {11 0} facesof a bcc truncated rhombic dodecahedron crystal can alsoinduce the formation of dendrite patterns with tips point-ing to the h100i directions [33]. Therefore, a unified regimefor fcc and bcc dendrite orientation selection processes isobtained by PFC simulation during the early stage of den-drite growth.

Our PFC simulation results reproduce the atomic orien-tation selection process, which helps to understand real fccand bcc dendrite orientation selection processes at the earlygrowth stage. As mentioned above, in two dimensions, pre-vious theoretical and experimental studies about snow den-drite growth [39–41] have shown the similar evolutionprocess from a polygon into a dendrite: the nucleation ofnew solid layers occurs near the corner of the polygon,and then grows up along the steps created by nucleationsimultaneously, and finally a dendrite pattern is obtainedin multi-layer growth mode. In the polygon crystal growth,the volume diffusion around the polygon results in non-uniform supersaturation distribution around the edges.The supersaturation is the highest at areas near the cornersbut decreases at areas near the center of the edges. There-fore, the area close to the corners provides the preferrednucleation position. In our simulation, the density diffusionaround fcc octahedron crystals also gives rise to the similarsupersaturation distribution around the {11 1} faces, caus-ing the preferred nucleation sites of new {111} layers atareas near the octahedron tips.The orientation selection

processes of fcc octahedron crystals and bcc truncatedrhombic dodecahedron crystals, as shown in our PFC sim-ulations, are helpful for understanding how the dendritesselect the tip direction in real systems, such as colloids, met-als, and in the electrochemical deposition of metals. ThePFC simulation has already successfully simulated the 2-D colloid dendrite growth with triangular lattice [43]. Forfcc colloid dendrite growth, although growth kinetics iswell investigated by previous experimental studies [44–47], the formation of tips and how the tips proceed in themelt have not been observed yet. Our simulated resultsfor the fcc dendrite tip’s formation and advancement areinstructive for the understanding of the colloidal dendritegrowth orientation selection process. The fcc real metalsusually have lower interface energy anisotropy than ourPFC fcc systems with an interface anisotropy ofe4 = 0.159. It is found from previous experiments andnumerical simulations that the fcc small nucleus tends toexhibit octahedral shape under small growth driving forcesduring the early stage of solidification [19,21,48]. More-over, the facet growth of the metal nucleus has beenobserved experimentally [49], especially at the initial stageof the crystal growth when the growth driving forces arelow. Therefore, although the magnitude of the surfaceenergy anisotropy adopted in our PFC simulations deviatesfrom the real fcc metal on the large side, our results for theorientation selection of facet dendrite growth are helpful tounderstand the orientation selection process of the real fccmetal dendrite at the early growth stage. During electro-chemical-deposition growth, a shape transition of the fccmetal nanoparticles from octahedron into dendrite patternshas already been verified by experiment [20]. However, sofar, it has not been possible to observe experimentallyhow this process performs. Our simulated results providea reasonable explanation of this process.

4. Conclusions

In summary, the orientation selection process during fccand bcc crystal growth is investigated using the PFCmethod at atomic scale with emphasis on the fcc dendritegrowth. Due to the crystalline anisotropy, the fcc octahe-dron crystal grows faster along the h100i directions thanalong other directions. The new layers on the fcc {11 1}plane tend to nucleate at the positions close to the tips,which drives the octahedron fcc crystal to lose its stabilityin multi-layer growth at large supersaturation, and furthercontributes to the formation of fcc dendrite patterns witharms pointing to the h10 0i direction. For bcc dendritegrowth, a similar orientation selection regime is obtainedduring the evolution from a truncated rhombic dodecahe-dron into a dendrite pattern. Our PFC simulated resultsrevealed the orientation selection process, being consistentwith the previous 2-D theoretical and experimental studiesabout snow dendrite growth, and help us to understand theorientation selection process of real dendrite growth.

S. Tang et al. / Acta Materialia 60 (2012) 5501–5507 5507

Acknowledgements

This work has been supported by the Nature ScienceFoundation of China (Grant Nos. 10974228 and51071128), the National Basic Research Program of China(Grant No. 2011CB610401), the fund of the State Key Lab-oratory of Solidification Processing in NWPU(SKLSP201001), Program for New Century Excellent Tal-ents in University, and the Doctorate Foundation ofNorthwestern Polytechnical University (CX201106).

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