Advances on Microstructure Modeling of Solidification Process of Shape Casting

TSINGHUA SCIENCE AND TECHNOLOGYISSN 1007-0214 01/18 pp497-505Volume 9，Number 5，October 200Advances on Microstructure Modeling of SolidificationProcess of Shape CastingLIU Baicheng (柳百成)* , XU Qingyan (许庆彦)Department of Mechanical Engineering, Tsinghua University, Beijing 100084, ChinaAbstract: Simulation technology for shape casting at macro-scale has been sccessfully put into engineer-ing application in a number of casting plants and as a result the quality of castings is assured, the researchand development time is shortened, and the manufacturing cost is greatly saved as well. In this paper, mod-eling and simulation technologies of soldification process of shape casting at microstructurescale, espe-cially deterministic, cellular automaton, and phase field models are studied and reviewed.Key words: microstructure modeling; deterministic model; stochastic model; Monte Carlo method; cellularautomaton; phase fieldweighing 62 tons for the Three Gorges Project wasIntroductionsuccessfully cast first at once in the Second HeavyCasting industry plays an important role to the devel-Manufacturing Group in 2001 by using advancedopment of national economy. Casting technology assimulation technology. It was reported that simulationwell as forging, welding technologes, ete. not only tchnology could guarantee the quality of castings,forms the machine component parts but also deter-shorten the research and development period, and thenmines the structure, the property, and the performancesave their cost as well. In this paper, progress and re-of these parts!"!. China has a long, brlianat casting his- view on micro modeling of sildification process oftory especially in bronze casting. The annual castingshape castings at microstructure-scale are presented.production has exceeded the United States and becomeMicrostructures are thermodynamically unstablethe largest cating production country in the world forms of matter aggregation. They evolve with timedue to the driving forces such as volumetric free en-research frontier, and modeling and simulation of so-lifcation of casting become one of the htest re- temporal microstucture evolution might be stimulatedsearch topics in both manufacturing and material sci-by extermal fields, such as temperature and fluid flowence fields. Nowadays, simulation technology espe-fields. In the past decade, microstructure simulationcially at macro scale has been extensively studied2-51has been paid much more attention. The approachesemerged for microstructure modeling can be generallycasting plants by using a number of commercial soft-classified into deterministic and stochastic approaches.ware such as MAGMA, PROCAST, and FT-STAR.Deterministic modeling is based on the solution of theFor example, the stainless water turbine blade castingcontinuum equations at macroscopic scale. Stochasticmodeling introduces the elements of randomness in theReceived: 2004-07-30simulation and can produce a graphic display similarSupported by the National Key Basic Research andwith real micro:-proachesDevelopment (973) Program of China (No. G20000672083)中国煤化工PTo whom crrespondence should be adressed.have been used:MHCNMHGpatiE-mail: liubc@ tsinghua.edu.cn; Tel: 86-10-62773514temporal microstrlYs of any498Tsinghua Science and Technology, Ocober 2004, 9(5): 497- 505sizes, while stochastic models are only applicable to evolution and on the final solidified microstructure wasrelatively small part of a whole casting because of theanalyzed.restriction of computer memory and computationalZhao and Liu'zused deterministic models tctime. Consequently, the hybrid models, which combine simulate the solidification and microstructure forma-the advantage of deterministic and stochastic models,tion of a real spheroidal graphite (S.G.) iron casting.have also been developed.The simulated results of the microstructure and me-chanical properties of the spheroidal graphite iron cast-1 Deterministic Modeling are shown in Fig.1.Based on solidification kinetics, deterministic mod-Thevoz et al!fl are the first ones to integrate a soluteelingo-l 3 has sound physical background for crystaldiffusion model at the scale of an equiaxed grain and agrowth. However, it did not consider the probabilisticdendrite growth kinetics model with a finite elementphenomena occurring in solidification and neglected(FE) solution of the heat flow equation. Theany aspect which is related to crystallographic effects,calculated average grain size maps and cooling curvessuch as the grain selection occuring close to the moldwith recalescence in castings. Their deterministicwall,“equiaxed-columnar" transition near to the moldmodels can only account for equiaxed grain structurewall, the variation of the transverse size of columnarand did not consider the transport by convection andgrains, and columnar-to-equiaxed transition in bulksedimentation.liquid.Wang and Beckermann',8l proposed to solve the en-ergy, species, and momentum conservation equations2 Stochastic Modelso as to consider the effect of solid transport on thepredicted equiaxed grain structures. Their micro modelOver the past decade, stochastic models have been de-considered three “phases” or regions, i.e., the solidveloped for the prediction of dendritic grain structuredendrite, the interdendritic liquid between the dendritesformation. For stochastic approaches, the probabilisticarms, and the extradendrtitic liquid around the grainmodel is used to describe the nucleation and growth ofenvelope. The main inovation is the possibility tograins. The main techniques presently used for prob-quantify the role of the solid and liquid transport on theabilistic modeling are the Monte Carlo technique andfinal grain structure. Nevertheless, their model cannotthe cellular automaton technique.account for the topology and texture evolution of typi-cal columnar zones.2.1 Monte Carlo modell4 16]Brown and Hunt'9 used the interface-tracking tech-nique on a fixed grid to determine the motion of theRealizing the limitation of deterministic models for so-dendritic front and the equiaxed grain boundaries.lidification, Spittle and Brownt developed a totallyMovement of massless front marker particles repre-different approach which is based on“probabilistic"sents the advance of the columnar front and equiaxedtheory. They introduced Monte Carlo method, whichgrain boundaries. Gandin' t0 developedone-was developed by Srolovitz et al. to treat the graindimensional solidification model for directionally castgrowth, to simulate the solidification. This method isingots, which is based on the solution of the heat flowbased upon minimization of the intrfacial energy, asequation using a two-interface front tracking techniqueachieved by considering the energy of unlike sites (e.g..accounting for the growth kinetics at the limits of theliquid/solid sites or sites belonging to different grains)mushy zone. A comprehensive grain-tracking modeland by allowing transitions between these states to 0C-was developed by Yang and Stefanescul " I for accuratecur according to randomly generated numbers. By us-prediction of micstrucure evolution of Al-baseda ing this method, Spitle and Brownh4 were able toloys. The model describes the effect of natural convec-produce two-dimensional computed microstructures,tion caused by the thermalsolutal buoyancy, the effectwhich resembled very closely those observed in realof sldifcation contraction, and the efeet of grain micrographic Cros_selectionmovement during solidification. The effect of the grainof grains in the c中国煤化Imartmovement during solidification on the microstructureequiaxed transitionMYHCNMHGiclybyLIU Baicheng (柳百成) et al : Advances on Microstructure Modeling of Solidification ...499mm~310-3 mm(a) Nodule counts%(b) Nodule diameterHB(C) Pearlite content(d) Brinell hardnessMPa(e) Tensile strength(I ElongationFig. 1 Simulated results of microstructure and mechanical properties of an S.G. iron castingusing this technique. These authors could also qualita- If the total density of grains nucleated at a given un-tively demonstrate the effects of solute concentration dercooling is obtained from an average distribution,or melt superheat upon the resultant microstructure.the locations of these sites are chosen randomly. TheHowever, as mentioned by these authors, Monte Carlo random crystallographic orientation of a newly nucle-method suffers considerably from the lack of physical ated grain is also taken into account in cellular automa-basis and cannot be used to analyze quantitatively theton model. The growth kinetics of the dendrite tips andeffect of the various physical phenomena. Furthermore,of the side branches is incorporated into the model inthe results are sensitive to the type of Monte Carlo such a way that the final simulated microstructure isnetwork used in computation and the calculation timeindependent of the cellular automaton network used forhas no obvious relationship with solidification time.the computation. The grain structure and size distribu-tion as well as the formation of columnar grain and the2.2 Cellular automaton modelcolumnar-to-equiaxed transition could be simulated byCombining the advantage of probabilistic methodsusing the cellular automaton method.with those of deterministic approaches so as to predictThe cellular automaton technique, originally devel-more accurately the grain structure in castings, cellularoped by Hesselbarth and Gobel4, was used first byautomaton is based upon the physical mechanism ofRappaz and Gandin'to the simulation of grainnucleation and growth of dendritic grains 20. Het-structure formation中国煤化工ncleation甘忙This tech-erogeneous nucleation was simulated by a nucleationnique consists isite distribution in deterministic solidification models.sites of random cYHCNMHGithinthe500Tsinghua Science and Techmology, October 2004, 9(5): 497- 505computational space, based on probabilistic relation-vector (computed from deterministic law) reachesships. Then deterministic laws, such as v=f(OT ), arethem. Some modifications on cellular automatonthen used to allow these nuclei to grow. The differentmodel were carried out.2. For example, the influ-micro-volume elements of the computational space canence of the cooling rate on the final grain structure ofonly be liquid or solid, and change state as the growthcastings is shown in Fig. 212” .3-D2-D0.50 K1.34K /s2.18K/sFig. 2 Influence of cooling rate on simulated microstructurDuring normal casting solidification process, a cast-in Fig. 3, the cross- section microstructure consists ofing solidifies gradually from outer to inner and often three zones: the chill zone composed of small equiaxedforms three microstructure zones: chill, columnar, andgrains, columnar zone with parallel columnar grainsequiaxed zones. At a certain time, the microstructurevertical to the mold side, and central equiaxed zonewill change from columnar to equiaxed, called colum- with coarsening grains. The columnar zone becomesnar-to-equiaxed transition, a very interesting phe- narrower while equiaxed zone becomes wider with thenomenon occurring in solidification process. The simu-decrease of pouring temperature and if the pouringlated results of microstructure variation with the differ-temperature decreases to a certain extent, the final mi-ent pouring temperature are shown in Fig. 3. As showncrostructure will consist of complete equiaxed.(a)Tp= 1823 K(b)7p= 1798K(C)Tp=1773 KFig. 3 Simulated results of CET at different pouring temperatureslandPang251 attemoted to modelmary stem, secondary arm, and high. order arms, which the dendritic morp中国煤化工cllularcannot be modeled by the previous cellular automatonautomaton technic| YHC N M H GographicLIU Baicheng (柳百成) et al : Advances on Microstructure Modeling of Solidification ....50anisotropy of the grains and the branching mechanismequilibrium interface temperature， including theof dendrite arms. In their micro -model, stochasticanisotropy of surface tension as well. The modelmodeling at a length scale of 10° m was coupled with reproduced qualitatively the parabolic shape of thedeterministic modeling at a length scale of 10* m. dendrite tip, the development of secondary and tertiaryArm coarsening was also calculated with a determinis-arms, the instability of the planar interface, thetic dendrite tip kinetics law and crystallographic con-coarsening driven by curvature reduction, and dendritesiderations in combination with a deterministic coars-selection under constrained conditions.ening model. However, the overall growth of dendriteNastac- ! ' proposed a more comprehensive stochas-arms was derived from probabilistic calculations. tic model for simulating the evolution of dendriticBranching of dendrites was allowed to occur based oncrystals during solidification. The model includes time-the classic criterion for morphological instability.dependent calculations for temperature ditribution,Dilthey and Pavlik5o) used a modified cellularsolute redistribution in the liquid and solid phases, cur-automaton model to simulate the grain morphology.vature and growth anisotropy. Simulation of grainThe solute diffusion was taken into account both ingrowth and dendritic morphology was also carried outliquid and solid as well as solute partition between and the details can be seen in Refs. [19, 20, 24]. Thethese two phases. The model incorporates the influencesimulated results of equiaxed dendritic grain are shownof solutal and curvature undercooling at theinFig. 4.CICT/K08(a) Equiaxed dendritic(b) Temperature distribution(C) Concentration distributiongrowth morphologyFig. 4 Modeling of equiaxed dendritic growthThe comparison between modeling and experimentalresults of columnar dendritic growths! is shown in Fig.5. The dendrite stems grow continually opposite to theheat transfer direction. At the same time, large quanti-ties of parallel secondary dendrite arms appear at twosides of the stems, which is close to the observed den-drite feature.3 Phase Field Model(a) Simulated branch growth (b) Ohserved branch growth of metal-Recently, phase field method49.301 is being paid morelike columnar dendritic growth insuccinonitrile- acton aloys啊and more attention. The structure of a solidliquid in-terface is determined by the competition between theFig. 5 Simulated and experimental competitive growthordering influence of structure and the thermal disor-of branches during columnar growthdering effects. Phase field theory reflects the influenceinterface in a metallic system. By coupling the phaseof diffusion, ordering potential, and thermodynamicfield equation to the temperature, solute, fluid velocity,driving force in terms of differential equations. Theand other external中国煤化工offers theresolution describes the shape, the curvature, and the prospect of being-umericalmotion of a diffusion interface, such as the solid/liquidsolidification expelMHCNMHG02Tsinghua Science and Technology, October 2004, 9(5): 497- 505Phase variable φ was introduced in phase field the-compared the experiments with phase field calculationory. φ is an ordering variable representing the physicalresults using an adaptive grid finite element procedurestate of the system (liquid or solid or interface). φ∈that tracks the moving solidification interface. Grafe et[-1, 1], where φ= 1 represents the solid phase and φ =al.5" " used a multi-phase-field model to predict micro--1 represents the liquid phase. The interface is definedstructure evolution and micro- segregation during direc-by the set of points defined byφ= 0.tional solidification of Ni-AI-Cr tertiary single crystalThe current researches on phase field include: 1)superalloy. The particle/interface problem was numeri-coupling between multiple phase fields in multi-graincally analyzed by Ode et al. [8] using phase field modelgrowth; 2) coupling to temperature or solute and den- in thin interface limit. The particle pushing and en-dritic growth; 3) coupling between two phase fieldsgulfment behaviors were successfully reproduced forand solute in peritectic and eutectic solidification; andthe system of Fe-C aloy and an alumina particle.4) coupling to velocity in a mush with forced convec-Zhang et al.59 extended phase field model of puretion. Nestler and wheeler'l presented a new formula-materials to binary alloys. The simulated dendritetion of a multi-phase field model, which includes bothmorphology of binary alloy compared with pure sub-surface energy and interfacial kinetic anisotropy. The stance is shown in Fig. 6.model was derived in a general form so that it has theflexibility to model and numerically simulate a varietyof solidification phenomena in peritectic, eutectic, andmonotectic alloys by suitably alternating the systemparameters. The relation between the free energy in themulti-phase-field model and the specific phase dia-gram of the alloy system was established by usingTiaden' 2 simulated a lamellar eutectic microstructure(a) Equiaxed dendritic grain of purefor a binary alloy with a symmetric phase diagram us-substanceing the multi-phase field. Diepers et al.3) studiednumerically the two-dimensional coarsening of abinary solid/liquid mixture to investigate the influenceof melt flow with a phase field method. In the limit ofpurely diffusive species transport, only minordeviations from classical LSW theory were observedthat could be atributed to coalescence. The forcedflow simulations revealed a faster ripening behaviorand a broader distribution radius of curvature, which(b) Equiaxed dendritie grain ofbinary aloywere in qualitative agreement with the Ratke-Fig.6 Dendritic growth simulation by using phase fieldThieringer theory of convection-influenced coarsening.Tong et al.!541 employed the phase field method toProvatas et al.40 studied the dendritic micro-investigate the interactions between melt flow and structure evolution using an adaptive grid, finite ele-deiriaingrgrwta. proposed a two-dimensional nu-ment method applied to a phase-field model. Figure 7merical model for solid/liquid interface to simulate theshows a dendrite 10 time steps into its evolution com-time-evolution of the perturbation at the interface. Theputed using the adaptive grid method. Side-branchingnumerical scheme was capable of predicting the dy-is evident, and arises due to numerical noise.namics of the spacing selection based on the localCha et al.41 developed a new phase field model togrowth velocity, interface velocity, interface tempera- study solidification of a multi-component alloy con-ture, and solute concentration at the inter face. Dantzigtaining substitutio| 中国煤化Irue ele.et al.50 summarized observations of dendritic growth ments. By emploCNMHCE perper unitunder both microgravity and terrestrial conditions andvolume as the cMHC:volutionLIU Baicheng (柳百成) et al : Advances on Microstructure Modeling of Soldification●.503Figure 9 shows several simulation snapshots takenby Loginova et al.421 that demonstrate the growth ofthree dendritic grains and their subsequent impinge-Temperaturement. Three seeds were introduced into an under-cooled melt. In the final picture, the secondary armshave coalesced within each grain, but liquid remains atthe grain boundaries because of the build-up of solutebetween grains and the energy penalty of forming theOrder pararneterInterfacegrain boundary.Fig. 7 A dendrite grown using the adaptive-gridmethod and phase field. Clockwise, beginning at theupper right the figures show contours of temperaturefield, contour φ = 0, contours of the (-field, and thecurrent meshequations of both the phase field and the concentration(fields were derived from the free energy functional inthe thermodynamically consistent way. Equiaxed den-drite shapes for Fe-0.5 at.% C binary and an Fe-0.5at.% C-0.005 at.% Mn tertiary alloy are pictured at theevolution time of 2.125x10-'s as shown in Fig. 8.(b(a) Mn concentration profile(eFig. 9 A phase-field simulation of alloy grainimpingementConm orMnObviously, phase field is very suitable for the directmodeling of dendritic grain morphology. However, it000is limited to a relatively small equiaxed grain, without1078a full side-branch structure, and relatively high-dimensionless undercooling. The extension to fullydendritic grains growing at the low undercooling oftypical casting processes would require an increase incomputational power by at least three orders of magni-(b) C concentration profiletude in both speed and memory.Fig. 8 Dendritic growth of Fe-0.005 at% Mn-0.5 at%A novel mesos; beenC alloy at 1775 K at evolution time of 2.125x 10~5s中国煤化工adeveloped by SteYHCN MH(5 theun-steady growth ofHc grains504Tsinghua Science and Techmology, October 2004, 9(5): 497- 505into an undercooled melt of a pure substance. In theConference on Advanced Materials and Processing, Hawaii,model, the numerical calculation of the temperatureUSA.2001, I and II: 2399 -2402.field in the undercooled melt between the grains is [5] Liu Baicheng, Xu Qingyan, Xiong Shoumei, Kang Jingwu.coupled with a stagnant-film model for dendrite tipProgress on multi-scale modeling on casting process. Joumalgrowth, such that without resolving individual dendriteof Mechanical Enginering, 2003, (10): 53-63. (in Chinese)arms the evolution of the grain envelope and the inter-6] Thevoz Ph, Desbiolles J L, Rappaz M. Modeling of equi-nal solid fraction can be predicted. However, the grainsaxed microstructure formation in casting. Metall. Trans,1989, 20A: 311-322.number involved in calculation is still limited. There-fore, the cellular automaton technique might be a good7] Wang C Y, Beckermann C. Multi- scale/phase modeling ofdendritic alloy solidification. American Society of Mechanicalway to describe the grain structure at mesoscopic scaleEngineers, Heat Transfer Division, (Publicaion) HTD, V 284,and dendritic growth and arm branching at microscopicTransport Phenomena in Soldification, ASME. NewYork,scale.USA, 1994: 75-95.4 Conclusions[8] Wang C Y, Beckermann C. Equiaxed dendritic solidifca-tion with convection: Part I. Muliscale/multiphase model-Modeling and simulation on solidification process ofing & Part II. Numerical simulations for an Al-4 wt% Cushape castings at macro-scale were extensively studiedalloy. Metallurgical and Materials Transactions A: Physi-and put into application in many casting plants incal Metallurgy and Materials Science, 1996， 27A(9):China, while modeling and simulation at microstruc-2754-2783.ture-scale become one of the hottest research topics in9] Brown D J, Hunt J D. A model of columnar growth using aboth manufacturing and material science fields.front-tracking technique. In: Modeling of Casting, WeldingCurrent situation and progress on deterministic, cel-nd Advanced Soldification Processes IX， Aachen,lular automaton, and phase-field models of micro-Germany. 2000, 8: 437-444.structure evolution of solidification process of casting[10] Gandin Ch-A. Fronduring directional solidification of dendritic alloy. In: Mode-were reviewed. Combining the advantage of probabil-ling of Casting, Welding and Advanced Solidification Proc-istic methods with those of deterministic approaches,esses IX, Aachen, Gemany. 2000 8: 473480.the cellular automaton method could be used to predict[11] Yang B J, Stefanescu D M. Solidification modeling withthe grain structure and size of castings. On the othergrain tracking. modeling of casting. In: Welding andAdvanced Solidification Processes IX, Aachen, Germany.diffusion, ordering potential, and thermodynamic driv-2000, 8: 582-589.ing force in terms of differential equations, and is suit-[12] Zhao Haidong. Numerical simulation of microstructureable to study in detail the dendrite growth duringformation and shrinkage defect prediction of spheroidalsolidification.graphite iron casting [Ph. D. Dissertation]. Beijing:ReferencesTsinghua University, 2000. (in Chinese)[13] Zhao H D, Liu B C. Modeling of stable and metastable[1] Liu B C, Jing T. Modeling and Simulation of Casting Pro-eutectic transformation of spheroidal graphite iron casting.cess and Quality Control. Beijing: Machinery IndustryISIJ Inter., 2001, 41(9): 986-991.Press, 2001. (in Chinese)[14] Spittle J A, Brown S G R. Computer simulation of the[2] Liu Baicheng. Progress in solidification modeling of castefects of alloy variables on the grain structures of cast-iron in China. International Joumal of Cast Metals Re-ings. Acta Metall, 1989, 37(7): 1803- 1810.search, 1999, 11(5): 259- 266.[15] Jiao Y N, Liu B C. Simulation of metal solidification using[3] Liu Baicheng, Kang Jinwu, Zhao Haidong. Study on macroa coupling macroscopic heat transfer and Monte Carloand micro modeling of solification process of shapedmethod. In: The 5th Asian Foundry Conference. Nanjing,casting. Chinese Journal of Mechanical Engineering, 2002,China, 1998, 5: 11-16.15(3): 252-256.[16] Zhu P, Smith R W. Dynamic simulation of crystal growth4] Liu B C, Xiong S M, Xu Q Y. Research on macro and mi-by Monte Carlo中国煤化工d kinetics.cro modeling of soldificationn process of shaped casting. In:Acta Metall. MPRIM 4: Proceedings of Fourth Pacific Rim International [17] Srolovitz D JYHCNMHGahniPs.LIU Baicheng (柳百成) et al : Advances on Microstructure Modeling of Soldification ..505Computer simulation of grain growth- H: Influence of a[33] Diepers H J, Beckermann C, Steinbach I. Modeling ofparticle dispersion. Acta Metall, 1984, 32: 1429-1438.convection-influenced coarsening of a binary alloy mush[18] Rappaz M, Gandin Ch-A. Probabilistic modeling of micro-using the phase-field method. In: Modeling of Casting,structure formation in solification process. Acta Metall.Welding and Advanced Soldification Processes VIIIMater, 1994, 41: 345-360.1998: 565-572.[19] XuQ Y, Liu B C. Modeling of as -cast microstructure of[34] Tong X, Beckermann C, Karma A. Phase field simula-Al alloy with a modified cellular automaton method. Ma-tion of dendritic growth with convection. In: Modelingter. Trans,, 2001, 41(7): 2316-2321.of Casting, welding and Advanced Solidification Proc-[20] Feng W M, Xu Q Y, Liu B C. Microstructure simulationesses VII. TMS, San Diego, USA, 1998: 613-620.of aluminum alloy using parallel computing technique.[35] Catalina A V, Sen S, Stefanescu D M, et al. NumericalISIJ International, 2002, 42(7): 702-707.modeling of the dynamics of the solid/liquid interface[21] Hesselbarth H W, Gobel I R. Simulation of rerysalliza-morphology during directional solidification of alloys. In:tion by cllular automata. Acta Metall. Mater, 1991, 39(9):Modeling of Casting, Welding and Advanced Soldifica-2135-2143.tion Processes IX, Aachen, Germany. 2000, 8: 445-452.[22] Gandin Ch-A, Rappaz M. Coupled finite element-cellular[36] Datzig J A, Provatas N, Goldenfeld N, et al. A comparisonautomation model for the prediction of dendritic grainof phase-field computation with experimental microgravitystructures in solification processes. Acta Metall, 1994, 42:2233- 2246.Modeling of Casting, Welding and Advanced Solidification[23] Shin Y H, Hong C P. Modeling of dendrite growth withProcesses IX, Aachen, Germany. 2000, 8: 453-460.convection using a modified cllular automaton model[37] CGrafe U, Botinger B, Tiaden J, et al. Multicomponent multi-with a dffuse phase. ISIJ Int., 2002, 42: 359-367.phase solidification of Ni based superalloys: Phase field simu-[24] Xu Qingyan, Liu Baicheng. 3-D stochastic modeling oflations coupled to thermodynamic database. In: Modeling ofgrain structure for aluminum alloy casting. Journal of Ma-Casting, Welding and Advanced Solidification Processes IX,terials Science and Technology, 2003, 19(5): 391-394.Aachen, Germany.2000, 8: 481 -488.[25] Stefanescu D M, Pang H. Modeling of casting solidifica- [38] Ode M, Lee J S, Suzuki T, et al. Numerical simulation of thetion: Stochastic or deterministic? Canadian Metallurgicalparicle/interface problem using a phase-field model. In:Quarterly, 1998. 37(3-4): 229-239.[26] Dilthey U, Pavlik V. Numerical simulation of dendriteProcesses IX, Aachen, Germany. 2000, 8: 521-527.morphology and grain growth with modifed cllular [39] Zhang G Y, Jing T, Liu B C. Microstructure simulation ofautomata. In: Modeling of Casting， Welding and Ad-aluminum alloy casting using phase field method. Int. J. .vanced Solidification Processes VII TMS, San Diego,Cast. Metal. Res., 2002, 15(3): 237-240.USA, 1998: 589-596.[40] Provatas Nikolas, Goldenfeld Nigel, Dantzig Jonathan. Ef-[27] Nastac L. Numerical modeling of solidification morpholo-ficient computation of dendritic microstructures usinggies and segregation patterns in cast dendritic alloys. Actaadaptive mesh refinement. Physical Review Letters, 1998,Mater, 1999, 47(17): 4253- 4262.80: 308-3311.1[28] Kurz W， Esaka H. Solidification microstructures. [41] ChaP R, Yeon DH, YoonJ K. A phase field model forPraktische Metallographie, 1988, 25(5): 207-213.isothermal solidification of multi-component alloys. Acta[29] Kobayashi R. Modeling and numerical simulation of den-Mater, 2001, 49: 3295-3307.dritic crystal growth. Plhys. D, 1993, 6: 410-423.[42] Bottinger W J, Warren J A, Beckermann C, Karma A.[30] Karma A, Rappel W J. Phase field method for computation-Phase-field simulation of solidification. Anual Review ofally eficient modeling of slidification with arbitrary inter-Materials Research, 2002, 32: 163- 194.face kinetics. Plhys. Rev. Lett, 1995, 53(4): 3017-3020.[43] Steinbach I, Beckermann C, Kauerauf B, et al. Three-[31] Nestler B, Wheeler A A. Anisotropic muti- phase-fielddimensional modeling of equiaxed dendritic growth on amodel: Interfaces and junctions. Phys. Rev. E, 1998, 57(3):mesoscopic scale. Acta Mater, 1999， 47(3): 971-9822603-2609.[32] Seeselberg M, Tiaden J. Simualtion of binary eutectic mi-crostructure using the muli-phase fieldl method. In: Mod-中国煤化工eling of Casting, Welding and Advanced SolidificationProcesses VII. TMS, San Diego, USA, 1998: 557-564.fHCNM HG