Dendritic Growth in Mg-Based Alloys: Phase-Field Simulations and Experimental Verification by X-ray...

Dendritic Growth in Mg-Based Alloys: Phase-Field Simulationsand Experimental Verification by X-ray SynchrotronTomography

MINGYUE WANG, YANJIN XU, QIWEI ZHENG, SUJUN WU, TAO JING,and NIKHILESH CHAWLA

Changes in polycrystalline dendritic growth patterns during solidification result in a variety ofsolidified dendritic structures and morphologies. These microstructural changes are induced bya variety of effects such as the random distribution of nucleation sites and orientations, theinteraction of growing individual dendritic grains, and effects of solid-liquid interfacial energyanisotropy. Here, we have studied the formation of the complicated and diverse dendritemorphologies both experimentally, by electron backscatter diffraction and by X-ray tomogra-phy; and numerically by three-dimensional phase-field simulations. Three binary magnesiumalloys were considered in this study: Mg-Al, Mg-Zn, and Mg-Sn alloys. We show that thesolidification microstructure can be attributed to the following factors: The interaction of thegrowing dendrites, the anisotropy of the growth, and the distribution and initial random ori-entations of nucleation sites.

DOI: 10.1007/s11661-014-2200-x� The Minerals, Metals & Materials Society and ASM International 2014

I. INTRODUCTION

DENDRITIC microstructures, which result from thedendritic growth of the solid-liquid interface (S/L), areubiquitous in a wide range of solidification processes,such as casting, welding, etc. The dendritic microstruc-ture, along with the inter-dendritic distribution andsegregation of elements and/or impurities, plays acrucial role in determining the uniformity and qualityof the final casting.[1–5] In addition to its technologicalimportance, dendritic solidification represents a ratherclassic example of self-organized formation in systemsfar from equilibrium. It is a phenomenon that occurs innature and has been studied extensively.[6–8] Neverthe-less, the quantitative understanding of dendritic evolu-tion is still a major theoretical and experimentalchallenge within the materials community.

Over the past few years, most of the research ondendritically solidified microstructures has been focused

on single crystals with well-defined crystallographicaxes. Studies on single crystals have provided anunderstanding of the orientation selection and evolutionof dendritic growth.[9–12] However, microstructures ofmost materials are polycrystalline three-dimensional(3D) structures. Solidification patterns and morpholo-gies in polycrystalline materials in three dimensions arequite complex and many fundamental questions remainunanswered.[13] The interactions between multiplegrains, which perturb the bulk diffusion behavior ofthe advancing arms of dendrites due to, for example,geometry and competing growth anisotropies, have aprofound impact on polycrystalline morphological evo-lution, yielding a wide spectrum of topologically com-plex dendritic patterns, especially in three dimensions.[14]

It is challenging to predict the complex microstructuresof alloys in 3D, especially when we take into account thedistribution, preferred directions of heterogeneousnucleation, and the wetting conditions for a grain onthe surface of preexisting nuclei or impurity particles.Actually, considerable observations on microstructuresof the industrial/commercial alloys indicate thatmost solidify into wide range of morphologies andtopologies.It has long been suggested that inherent crystalline

anisotropy plays a critical role in the evolution ofsolidified microstructures, especially in the context ofdendrites.[5,9,15,16] Historically, the orientation-relatedanisotropy in face-centered cubic (FCC) and/or hexag-onal close-packed (HCP) metals has been described withjust a single anisotropy term. More recently, however,atomistic-scale calculations, such as molecular dynam-ics, have demonstrated that accurate parameterizationsof the S/L free energy for FCC and HCP metalsgenerally require two or more anisotropy parameters

MINGYUE WANG, formerly Ph.D. Candidate with the School ofMaterials Science and Engineering, Tsinghua University, Beijing100084, P.R. China, and Visiting Scientist with the Materials Scienceand Engineering, Arizona State University, Tempe, AZ 85287, is nowPostdoctoral Fellow with the International Research Institute forMultidisciplinary Science, Beihang University, Beijing 100191,P.R. China. YANJIN XU, Postdoctoral Fellow, is with BeijingAeronautical Manufacturing Technology Research Institute, Beijing100024, P.R. China. QIWEI ZHENG, Ph.D. Candidate, and TAOJING, Professor, are with the School of Materials Science andEngineering, Tsinghua University. SUJUN WU, Professor, is with theInternational Research Institute for Multidisciplinary Science, Bei-hang University. NIKHILESH CHAWLA, Fulton Professor ofMaterials Science and Engineering, is with the Materials Scienceand Engineering, Arizona State University. Contact e-mail:[email protected]

Manuscript submitted October 1, 2013.

METALLURGICAL AND MATERIALS TRANSACTIONS A

associated with both four- and six-fold anisotropy andadditional terms in 3D space.[5]

An important class of lightweight metallic alloys,where knowledge of dendritic growth is important, ismagnesium alloys. Many commercially available mag-nesium alloys still do not satisfy some requirements formechanical properties, e.g., in automotive or aerospaceapplications, especially at higher temperatures.[17,18] InMg-rich alloy systems, the grain size and the dendriticmorphology of the primary HCPMg phase have a directeffect on the composition and distribution of micro-structures and, thus, on the properties of materials. Animportant characteristic of the microstructures in Mg-rich alloys is that the primary matrix phase leads to adifferent dendrite morphology compared to the typicalcubic dendrites observed in other alloys with FCCphases.

Processing conditions, i.e., analogously directional orequiaxed solidification fashion, can concurrently andextrinsically affect the evolution of microstructure andanisotropy, and they are also a direct footprint ofmicrostructural morphologies. The correlated effect ofvarious anisotropies intrinsically and extrinsically,therefore, magnifies the selection space of the classicminimum stiffness criterion commonly assumed todefine crystal growth directions, leading to the emer-gence of extremely complicated dendritic patterns.

A very recent study by Amoorezaei et al.[19] andGurevich et al.[20] showed that Mg-0.5 wt pct Al alloy’ssurface tension anisotropy can be correlated withextrinsic anisotropies present in processing conditionsduring solidification, which suggested that it is possibleto produce a continuous transition, and hence a morewide spectrum of morphological diversity from dendriticto seaweed and fractal-like structures by varying theprocessing conditions.

Herein, we deal with some non-trivial questionspresented above by combining polycrystalline phase-field algorithm which consider anisotropic effects, andX-ray synchrotron tomography to experimentally verifyanisotropy-induced morphological changes. Experimen-tally, to date, few detailed investigations have beenconducted on the 3D morphologies of a-Mg dendrites inMg-Al (HCP-FCC), Mg-Zn (HCP-HCP), and Mg-Sn(HCP-BCT) alloy systems. Directly measuring a-Mgdendrites in 3D is an important part of this work. Thedendritic crystal orientations of a-Mg were examined byEBSD. 3D morphological features of a-Mg wereexplored in three binary Mg-based alloy cases, wheredirectional and equiaxed solidifications were employedto confer the diversity of morphologies. The release ofheat was controlled to a very slow rate to insure aminimal effect of the interfacial kinetics and the fulldevelopment of the dendrite. Numerically, rather thanusing well-known cases of cubic alloy systems such asAl- or Ni-based alloys which have well-accepted aniso-tropic S/L functions, we instead choose Mg-based alloyswith hexagonal symmetry, which confer a more complexuncertainty for the anisotropy model, and hence give amore challenging and stringent test for this generalmechanism of multiple-grain morphological phenom-ena. Our work suggests that the diverse morphologies

arising from the combined effect of anisotropic growth,initial random orientations, and interactions of advanc-ing individual grains exist more widely than previouslythought. Strictly speaking, the complex interactions ofnumerous factors, such as buoyancy and flow ofdendritic free grains etc., make it difficult to accuratelypredict, geometrically, certain specific morphologies.

II. PHASE-FIELD MODEL

In this section, we present the details of a polycrys-talline algorithm for dendritic solidification in 3D. Inthis algorithm, in order to describe the orientationalrelationships from grain-to-grain, a modified localorientation field is proposed to inherently be incorpo-rated into the phase-field model. The modified localorientation field is used to deal with grain boundariesdirectly, while the preferred growth directions of indi-vidual dendrites are controlled by the anisotropic S/Lmodel. It has been proven to be a feasible approach toresolve the complexity of dendritic polycrystalline solid-ification in 3D.[14,21–23]

A. The Theory of Phase-Field Model

Phase-field models derived from both the local equi-librium and the variation principle, proposed decadesago to address the difficult problem of tracking thecrystalline interface, have been adopted to predict thevarious complex solidification patterns, especially den-drites.[5,9,14] The primary idea behind phase-field modelsis to replace the singular macroscopic treatment of aninterface by a regularized description through theintroduction of a phase-field variable, u. Since compre-hensive reviews exist on this topic,[24–27] detailed deri-vations of the phase-field equations and numericalsimulation techniques are not discussed here.In our model, the total free energy of the two-phase

system is described by a phenomenological Ginzburg-Landau model as:

F ¼Z

V

f u; uð Þ � c2 nð Þ2ruj j2

� �dV; ½1�

where F is the total free energy of the system, f(u, u) isthe free energy density function, u is the phase-fieldvariable parameter ranging from negative one in theliquid to positive one in the solid, u is the dimension-less temperature, defined as u = (T � TM)/(L/CP), andc(n) is the gradient energy coefficient. The governingequations based on thermodynamic theory for thephase-field coupled with temperature field can then beexpressed as follows:

l nð Þ @u@t¼ � dF

du; ½2�

@u

@t¼ Dur2uþ L

@h uð Þ@t

; ½3�


where l(n) is the kinetic coefficient, h(u) is the solidfraction given by u2(3 � 2u), Du is the thermal diffusioncoefficient, and L is the latent heat.

It should be noted that there is a significant amount offreedom in the choice of the functional form of the freeenergy density f(u, u) in the phase-field model. It shouldsatisfy the main thermodynamics requirements of: (i) theform of a double well potential with two minimacorresponding to the liquid and solid phases; and (ii)the difference in bulk free energies between solid andliquid, which is the driving force of the phase transfor-mation, should be a monotonically increasing functionof the degree of interface undercooling. A choice, whichsatisfies the above requirements, is proposed by Karmaet al.[28] as f u; uð Þ ¼ �u2=2þ u4=4

� �þ kuðu� 2u3=3þ

u5=5Þ. This form has been widely used to modeldendritic solidification quantitatively and has beendemonstrated to work well.

B. The Algorithmic Description for DendriticSolidification of Polycrystals

Warren et al.[21] and Kobayashi et al.,[22] and Granasyet al.[14] and Pusztai et al.[23] presented a phase-fieldmodel for the simulation of polycrystalline growth in 2Dand 3D, respectively. In their model, the four symmetricEuler parameters are adopted to define crystallographicorientations in three dimensions, allowing the descrip-tion of the 3D polycrystalline anisotropy with variouscrystallographic orientations. The model is capable ofreproducing various polycrystalline solidification pro-cesses including impinging dendritic particles, branchingneedle crystals, and spherulitic polycrystals in threedimensions.

To be applicable to an arbitrary number of differentphases or grains of the same phase, but distinct by theirorientation, the so-called multi-phase field model (MPF)using multi-variables of phase fields was developed bySteinbach,[27] Nestler et al.,[29,30] Eiken et al.,[31] Hechtet al.,[32] and MICRESS.[33] Each grain is distinguishedfrom others by its orientation of phase (or both) by itsindividual phase-field variable. Historically, this can beseen as a vector-order-parameter model in Landau’ssense.

In the present paper, in order to deal with differentcrystallographic orientations during polycrystallinesolidification, local rotation coordinates were assignedto every grain. Thus, any individual grain has an initialcrystallographic orientation of its own in the solidifi-cation process. As demonstrated in Figure 1, a randomorientation of dendritic grain i was obtained by ithlocal rotation coordinate specific to dendrite i. Thismeans that the local crystallographic orientation isconsidered as the relative orientation of a localcoordinate system fixed to the specific grain, basedon a general reference system for the whole computingregion. The relative orientation from grain-to-grain isuniquely defined by a single rotation matrix expressedin terms of the three Euler angles, as shown in Eqs.[4a], [4b] and Figure 1.

Rotate1 ¼coswi sinwi 0

sinwi � coswi 0

0 0 1

0B@

1CA;

Rotate2 ¼cosui 0 sinui

0 1 0

� sinui 0 cosui

0B@

1CA;

Rotate3 ¼1 0 0

0 cos hi sin hi0 � sin hi cos hi

0B@

1CA

½4a�

RotateMatrixð Þi ¼Rotate1 �Rotate2 �Rotate3;

xi

yi

zi

0BB@

1CCA¼ RotateMatrixð Þi�

xo

yo

zo

0BB@

1CCA½4b�

C. The Description of Anisotropic S/L of HCP in 3D

Over the past few decades, the fundamental andsignificant understanding of the mechanisms for dendriticorientation selection has been achieved.[1,3–5,12,16,19,20,34–39]

With the wealth of molecular dynamics and phase-fieldsimulations, as well as a few limited set of experiments,some interesting insights into the dendritic growth direc-tions, which are extremely sensitive to weak multi-termanisotropies of S/L free energies and mobilities, have beenachieved. It has long been assumed that dendritic direc-

Fig. 1—The schematic diagram of local selection growth orientationof dendritic grain i by using a rotation matrix of local coordinatesystem fixed to that specific grain.


tions are primarily determined by the anisotropy of S/Lfree energy, c(n), and more precisely, by the minima ofstiffness—a quantity of the sum of c and c¢¢ with respect toorientation angle h, that is, Sij ¼ csl þ c00sl ¼ ðcdij þ @2c=@/i@/jÞ, where dij = 1 for i = j and dij = 0 for i „ j, /i

and /j represent deviations of two orthogonal directionsfrom normal vector. Thus, one can define the growth traceby the following equation:

Trace S ¼ 2cþ @2c

@h2þ 1

sin2 h� @

2c

@/2þ cot h � @c

@h; ½5�

where h and / represent angular variables in sphericalcoordinates. We, thus, use a hexagonal function[40–44]

of Eq. [6] to descriptively show the effect of anisotropyon the dendritic grain morphologies of the equilibriumstate, as shown in Figure 2.

cS=L h;/ð Þ ¼ c0S=L 1þ e20y20 h;/ð Þ þ e40y40 h;/ð Þ½þ e60y60 h;/ð Þ þ e66y66 h;/ð Þ þ . . .�:

½6�

In order to reflect two types of preferred orientationalgrowth underlying hexagonal symmetry Mg, anisotropicfunctions of interfacial free energy based on Eq. [6] wereused in the paper to yield the 3D S/L interfacial patternsas expressed in Eqs. [7a] and [7b].

cðnÞ ¼ c0 1þ e1 n6x � 15n4xn2y þ 15n2xn

4y � n6y

� �þ e2n

6z

h i;

½7a�

cðnÞ ¼ c0 1þ e1 n6x � 15n4xn2y þ 15n2xn

4y � n6y

� �h

þ e2ð5n4z � 5n2z þ n6zÞ�;

½7b�

Fig. 2—The parameterized effect in anisotropic function on dendritic grain morphologies of equilibrium state. (a) e20 = �0.0082, e40 = 0,e60 = �0.05. (b) e20 = �0.0082, e40 = 0.05, e60 = �0.05. (c) e20 = 0.0082, e40 = 0.0, e60 = 0, e66 = �0.00205. (d) e20 = �0.3236,e40 = 0.0206, e60 = 0, e66 = �0.02733 (Color figure online).


where ni is an unit vector and is given by

ni ¼ @u@i

, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@u@x

� �2þ @u

@y

� �2þ @u

@z

� �2rand i represents x, y,

z, respectively, c0 is the mean values of the interfacefree energy, and e1 and e2 are the first- and second-or-der anisotropic parameters of crystal-melt interfacefree energy, respectively. Because of the lack of reliabledata on the magnitude of the anisotropy of the S/Lfree energy for magnesium alloys, c0, e1, e2 etc. neededto be chosen more pragmatically and cautiously. Spe-cifically, they were empirically selected based on previ-ous simulations.[36,37] The phase-field equation of Eq.[2] can then be rewritten by inserting the free energydensity function and the anisotropic factor in the fol-lowing form

lðnÞ @u@t¼ 2u� u3 � u5� �

� ku 1� u2 � u4 þ u6� � �

þr c2 nð Þru �

þ @

@pruj j2c nð Þ @c nð Þ

@ @u=@pð Þ

� �;

p ¼ x; y; z: ½8�

III. THE DISTRIBUTION AND INITIALORIENTATION OF HETEROGENEOUS

NUCLEATION

Almost all molten metallic alloys contain a number offoreign particulates or chemical heterogeneities of suf-ficiently small sizes that can act either as effectivenucleation catalysts or as substrates for new phases’precipitation. These particles provide a random resourcefor heterogeneous nucleation, together with container orcrucible walls. Actually, they also play a critical role inperturbing bulk diffusion behavior during solidification.

In this work, we consider the influence of distributionand initial orientation of heterogeneous nucleation inspace on complex dendritically polycrystalline solidifica-tion microstructures by coupling the interaction ofanisotropies in simulations. Figure 3 demonstrates theeffect of different distributions of foreign and initialorientations of wetting during heterogeneous nucleationon growthmorphologies. Indeed, these come into play bythe particularly initial orientations and combined growthanisotropies of S/L and interactions of each other.

IV. EXPERIMENTAL PROCEDURE

Mg-9 wt pct Al, Mg-33 wt pct Sn, and Mg-40 wt pctZn cast ingots were obtained by melting Mg (99.95 pctpure), Al and Sn (99.9999 pct), and Zn (99.999 pctpure), respectively, in an electrical resistance furnace at~1023 K (~750 �C) and poured into a cylindrical graph-ite crucible with a protective CO2-SF6 flowing mixturegas over the metallic liquid. Samples were cut from theas-cast ingots, remelted in sealed steel tubes (5 mm indiameter and 5 cm in length) with a BN coated andre-solidified in equiaxed mode with a very slow cooling

rate of 273.1 K s�1 (0.1 �C s�1) in a temperature-controlledfurnace. The sample tubes were taken out quickly andthen quenched in ice-water at desired temperatures to‘‘freeze’’ the microstructures. The quenching tempera-ture was set to 283 K (10 �C) above the correspondingeutectic temperature [e.g., 624 K (351 �C) for Mg-Zn] toform a solid-liquid mixture consisting of a small volumeof a-Mg dendrites and frozen Mg-X matrix melts. In thisstudy, Mg-Zn and Mg-Sn alloy composition was judi-ciously selected to be slightly different from the eutecticconcentration (such as CE = 51 wt pct Zn for Mg-Zn).This composition facilitated the experiments in twoways: (a) it provided a fairly small solid fraction at theend of the solidification process so that well-developeddendrites were not connected; (b) it also insured relativelarge compositional difference between the dendrite andthe melt, so that good contrast between the phases couldbe obtained during X-ray tomography.[37,38,45]

For the EBSD analysis, the samples were cut andpolished with SiC paper using ethanol as a lubricant.Electropolishing was then performed using a stainlesssteel cathode at a temperature between 263 K (�10 �C)and 253 K (�20 �C), for 40 to 60 seconds, with a DCvoltage of 9 V, and a current density of ~0.2 to0.3 A cm�2. Electropolishing was conducted in anelectrolytic solution of 5 vol. pct perchloric acid inethanol. The EBSD analysis was performed on a FEIQuanta 200FEG scanning electron microscope (30 kV,spot size of 1.2 to 3 nm, working distance between 10and 30 mm). The indexing of the pseudo-Kikuchi lineswas performed with the TSL OIM� 5.31 AnalysisSystem from EDAX Inc.X-ray microtomography measurements were carried

out at the BL13W1 beamline of the Shanghai Synchro-tron Radiation Facility (SSRF) (Figure 4). This beam-line offers fast acquisition of tomographic data at

Fig. 3—Schematic of the effect of particles distribution and nucleiinitial orientations during heterogeneous nucleation of bulk under-cooling liquid sea.


submicron spatial resolution. A tunable monochromaticX-ray energy of ~36 keV was used to penetrate acylindrical Mg-X sample of approximately 1.2 mm indiameter, 2 to 5 mm in height, with a voxel size ofapproximately (0.74 lm)3. A YAG:Ce scintillator screenwas used to convert the transmitted X-rays to visiblelight. This was coupled with a 2048 9 2048 pixels withan Optique Peter CCD camera with typical exposuretimes between 80 and 2000 ms per projection. In thisconfiguration, we collected a projection every 1/5 or

1/8 deg between 0 and 180 deg using an acquisition timeof 550 ms per projection. The 2D projections werereconstructed to slices of 3D using a filtered-back-projection algorithm with the software PITRE. Theresulting serial sections were saved as series of 32-bitTIFF files and were used to create 3D computer modelsof the objects by a serial of image processing andreconstructing the connected surfaces. They are cut andthe resulting matrix data size was~1800 9 1800 9 1500 voxels with a voxel size of(0.74 lm)3. The data set was segmented using thealgorithm of local dynamic growth, instead of overallthresholding, through Mimics 14.1 software (Material-ise).[37]

V. RESULTS AND DISCUSSION

From 3D X-ray computed tomography experimentsfor Mg-Al,[36,37,45] Mg-Zn,[38] Mg-Sn alloys, as shown inFigures 5 and 6, we first found that a-Mg dendritesshowed diverse morphologies when the alloy systemsand the processing conditions change. For example, inMg-Al alloys, the a-Mg dendrite has a typical plate-likestructure,[37] while it presents more complicated mor-phologies in Mg-Zn[38] and Mg-Sn alloys. These resultsinspired the following questions: What causes thediversity of morphology structure and what is theunderlying mechanism of the phenomenon? How does

Fig. 4—Experimental setup of BL13W1 beamline in SSRF used forX-ray tomography measurement.

Fig. 5—3D reconstructions of complicated morphologies of a-Mg dendrites in Mg-40 wt pct Zn alloys.


one, at least partially, predict these complex microstruc-tures using phase-field modeling?

It is possible that coarsening of early nucleated grainsmay have taken place, as shown in Figure 5(c). Inaddition, buoyancy effects may be possible in view of alarge density difference of a-Mg dendrites and thematrix. We have not taken the buoyancy and coarseningeffects into account in present phase-field simulationsdue to a quantitative difficulty of incorporating of thebuoyancy effect. We expect the buoyancy effect to beminimal within the time ranges involved. In order tocalibrate the anisotropic function for phase-field simu-lations, the orientations for Mg alloys were determinedusing electron backscatter diffraction (EBSD). We alsoespecially noted previous researchers’ measurementsand/or MD computation for Mg,[40] Zn,[46] and Sn.[47]

Al, for example, has a weak S/L anisotropy of ~0.01.[48–51]Sun et al.[40] computed the anisotropic data for pureMg as follows: (c1120 � c1010)/2c0 � 0.18 ± 0.08 (pct);

(c1120 � c0001)/2c0 � 1.2 ± 0.7 (pct), which showed theevident preferred orientation of h1120i in (0001) basalplane. Miller and Chadwick[46] also point out that theanisotropy of Mg is relative high via the measurement ofthe quenched equilibrium shape of liquid droplets in asolid matrix. For Zinc, Rheme et al.,[49] Mariaux andRappaz,[50] Rappaz et al.[51] used values of 70 mJ m�2 of(0001) and 100 mJ m�2 of (1010) etc. to compute the Al-Zn and Zn-Al solidified morphologies by phase-fieldmethod. The backscattered electron micrographs of aMg-Al alloy and a Mg-Zn alloy from experiments areshown in Figure 7. In our previous study,[37] plate-likea-Mg dendrite was shown to have a preferred h1120iorientation. In addition, a-Mg(Zn) dendrites formed inthe Mg-Zn melts showed two preferred orientations.[38]

So, we hypothesize that, together with other’s researchmentioned above and considering the uncertainty of theS/L anisotropy of Sn, the Mg-Al alloy has only onepreferred dendrite growth orientation, i.e., h1120i,[36,37]

Fig. 6—X-ray tomography original diagram slices of Mg-33 wt pct Sn showing that a-Mg morphology appears to a certain extent the curlingand disorder features. All micrographs used the same scale bar.


while the Mg-Zn alloy is believed to have two preferredorientations of h1120i and h2245i.[38] Accordingly, twoanisotropic functions for Mg-based alloys can thereforebe determined in Eqs. [6] and [7.1], [7.2].

We performed simulations for two distinct aniso-tropic growths of polycrystals using phase-field modelaccounting for the thermal noise. Simulations for boththe equiaxed and directional solidifications were carriedout. The non-dimensionalized physical properties andcomputational parameters used in the simulation for

Mg-Al and Mg-Zn systems are listed in Table I. Notethat this polydendritic grain growth algorithm is notrestricted to any alloy systems. 3D benchmark simula-tions of single dendritic growth shown in Figure 8,however, were carried out first in order to schematicallydemonstrate the remarkably anisotropic effect of S/Lgrowth on resulting morphologies. Note that we selectedan alternative combination of two preferred orientationsof h1120i and h0001i, which differs from the previouspolycrystalline growth simulations.

Fig. 7—Backscattered electron micrograph and corresponding pole figure: (a) a-Mg phase in the Mg-9 wt pct Al; (b-c) a-Mg dendrite formedfrom a Mg-40 wt pct Zn melt (Color figure online).

Table I. The Selected Non-dimensionalized Parameters Used in the Simulation

Alloys

Dimensionless Parameters

Du c0 e1 e2 l0 Du k Dx Dt

Mg-Al 0.49 0.35 �0.16 0.06 1 �0.65 9 0.4 0.01Mg-Zn 0.6 0.44 �0.1 0.03 1 �0.45 6 0.4 0.01

Fig. 8—The anisotropic effect of solid-liquid interfacial energy on the solidified dendritic morphologies with the phase-field simulations of singledendritic grain: (a) One preferred orientation of h1120i; (b) The combination of two preferred orientations of h1120i and h0001i.


Figures 9 and 10 show single dendritic grain mor-phologies extracted from the multigrain phase-fieldsimulation results. Two modes of solidification with

ten randomly oriented dendrites using one preferredorientation of h1120i (Figures 11(a) and (c)) and thecombined h1120i and h2245i (Figures 11(b) and (d))

Fig. 9—Complicated single dendritic grain morphologies extracted from multigrain phase-field simulations of randomly oriented ten dendritesusing one preferred orientation of h1120i: (I) equiaxed solidification with 501 9 501 9 501 grids vs (II) directional solidification with701 9 701 9 301 grids.


Fig. 10—Complicated single dendritic grain morphologies extracted from multigrain phase-field simulations of randomly oriented ten dendritesusing two combined preferred orientations of h1120i and h2245i: (I) equiaxed solidification with 501 9 501 9 501 grids vs (II) directional solidifi-cation with 701 9 701 9 301 grids.


favored orientations are shown: (I) equiaxed with501 9 501 9 501 grids and (II) directional solidificationwith 701 9 701 9 301 grids. The appearance of thedendrites of the single grain from polycrystalline simu-lations and the experiments (Figure 5) has a strikingsimilarity.[38] In the present phase-field simulations, wejust consider the combined effect of growth anisotropyand randomized distribution and initial orientations ofnucleation sites and an interaction of each other, whichstands as a sharper test of this simple mechanism bywhich morphologically diverse multigrain dendrites canbe well reproduced. From Figure 6, we also noted thata-Mg dendrites formed in the Mg-33 wt pct Sn, to acertain extent by the formation of disordered dendrites.

Granasy et al.[14] have used the phase-field method tostudy the growth of thin films in clay-polymer PEO/PMMA blends. They put forward a general polycrys-talline morphology formation mechanism. They claimedthat the particles additives, in addition to serving asheterogeneous nucleating sites, played a critical role indisrupting the process of crystal growth, leading to awide range of irregular morphology formation. Deflec-tion of advancing dendrite tips by engulfing the partic-ulates yields a variety of polycrystalline morphologies,and the effect will be more severe especially when thenumber and the density of randomly distributed heter-ogeneities are higher.

It is noted that the characteristics of a polymer fluidare very different from those of metallic alloys melt interms of physical properties, such as viscosity andadvancing dendrite stiffness. Unlike polymers, it is hardto imagine that the heterogeneous particles, on the onehand, work as obstacles blocking the grain growth, oron the other hand, function as catalyst for breakingsymmetry by rotating itself to match the optimumalignment of the dendrite/melt interfaces Furthermore,Granasy et al. focused on the film growth of thickness ofabout 17 nm, which is so thin that the influence ofheterogeneous particles is significantly weakened oncrystal growth morphology. It almost can be consideredas a 2D problem which may not be applicable in 3D.Actually, these contradictory effects of heterogeneousparticle present more similarity to the thermodynamicfactors than kinetic factors. Of course, as a consequenceof disturbing crystal growth, heterogeneous particlesgrowth front nucleation (GFN) may better explainthe diverse crystal morphology seen in polymers.Glicksman[4] mentioned that dendrite tips splittingmay make contributions to spontaneous symmetrybreaking, resulting in the appearance of disorderedstructures. Singer et al.[12,52–54] observed that phenom-enon during crystal growth of Xenon, which has similarphysical properties to metallic alloys melts. Parabolic,double, and triple tips can form depending on the degreeof undercooling.

As mentioned above, however, the metallic alloymelts are different. In fact, the crystal growth of a-Mg inthe different Mg-X melts system (i.e., HCP-FCC, HCP-HCP, and HCP-BCT) in our research is undertaken in acondition of a small degree of undercooling. Thermo-dynamics factors therefore dominate and the kineticfactors can be ignored. It is believed that the diversity of

polycrystalline solidification morphologies can be attrib-uted to the following factors: The distribution, inher-ently preferred growth directions, number and shape ofheterogeneous nucleation, wetting conferring variablyinitial orientations, the complex and varying anisotro-pies of a-Mg, and the interaction of individual grains.Figure 9 demonstrates complicated single dendritic

grain morphologies extracted from the multigrainphase-field simulations of randomly oriented ten den-drites using one preferred orientation of h1120i. The leftside of the figure presents results for equiaxed solidifi-cation with 501 9 501 9 501 grids, and the right side ofthe figure shows results for directional solidification with701 9 701 9 301 grids. Figure 10 represents the simu-lation results considering two favored growth orienta-tions h1120i and h2245i. So one can directly putFigures 9 and 10 together to estimate the anisotropicgrowth effect of S/L on crystal morphology undercertain condition of randomized nuclei’ distribution andinitial orientations of a multigrain in a confined geom-etry.This paper used the phase-field simulation inherently

considering the interplay of the above three factors totest an origin of morphologically diversity of polygraindendritic solidification. Although our simulation resultsof Figures 8, 9, 10, and 11 and 3D experimental resultsare not entirely identical, they are essentially

Fig. 11—The whole polycrystalline morphologies of phase-fieldsimulations using alternative preferred orientation.


self-consistent considering the randomness and theinteraction in actual solidification process and absenceof setting the associated parameters in the simulation.These complex interactions indeed produce many kindsof dendritic morphologies undetermined by simulationsof single grain growth.

VI. CONCLUSIONS

In summary, this work shows that complicatedpolycrystalline morphologies occurring in the solidifica-tion process can be reproduced in a confined geometryby phase-field simulations considering the synergeticeffects of varying anisotropic growth, initial nucleationorientations, and interactions of individual dendriticgrains, especially in view of the influence of S/L interfaceanisotropies on the morphologies. It has been shownthat the microstructural diversity can be attributed tothe following factors: The interaction of the growingdendrites, the anisotropy of the growth, and the initialrandom orientation of the nucleation sites. The resultsexplained the ubiquity of a great topological variety ofsolidified multigrain morphology in many polycrystal-line systems undergoing dendritically solidification pro-cess, regardless of their geometry. In one sense, it mayopen a route for controlling solidification microstruc-tures by manipulating the interaction of bulk diffusionfield, controlling initial nucleation orientations, adjust-ing the preferred growth directions, and using specificgeometries for solidification.

ACKNOWLEDGMENTS

MYW and TJ acknowledge financial support fromthe National Science Foundation of China, underGrant No. 51175292, Doctoral Fund of Ministry ofEducation of China, under Grant No. 20090002110031,and National Science and Technology Major Project ofChina, under Grant No. 2011ZX04014-052. MYW alsogratefully acknowledges the use of X-ray synchrotronbeam line BL13W1 at the Shanghai Synchrotron Radi-ation Facility (SSRF) and the Chinese ScholarshipCouncil for financial support during his stay at ASU.

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Dendritic Growth in Mg-Based Alloys: Phase-Field Simulations and Experimental Verification by X-ray...

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