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1、Alpha-Mg树枝晶三维形貌研究I：相场建模 Study of three-dimensional morphology of -Mg dendrite I:Phase-field modeling#5101520253035WANG Mingyue, SHUAI Sansan, JING Tao* Department of Mechanical Engineering, Tsinghua University, Beijing 100084 Abstract: The main objective of the present study is to establish numerica
2、lly a fundamentalunderstanding of the three-dimensional 3D dendritic morphologies of primary alpha-Mg solidsolution phase using phase-field modeling. An expression is proposed for the anisotropic function ofcrystal-melt interfacial free energy for hexagonal metals, based on the combination of experi
3、ments andcrystal structure. The phase field model of alloys, whose density free energy is built on the basis ofthermodynamic extended substitutional-regular-solution approximation, incorporated into anisotropicfunction reflecting hexagonal symmetry, is established. The governing equations for a dilu
4、te binaryalloy are derived, and numerical computations are implemented. The three-dimensional dendriticmorphologies of magnesium alloys microstructures, whose hierarchical branches can be seen clearly,are obtained, which have been well in agreement with experimental results and are obviously differe
5、ntfrom that of would be usually expected.Key words: phase-field;solidification; alpha-Mg; three-dimensioanl morphologies; dendrite0 IntroductionThe nature of solidified microstructures, with the following processes, controls the properties andqualities of the final product 1. One of the most widely
6、studied solidification microstructures, and alsoone of the most aesthetically pleasing, is the dendrite, with the best known example being thesnowflake. The critical role of dendritic growth in many important natural and technological processesmakes understanding the mechanism of hierarchical branch
7、es a high-priority research endeavor 2-4.The nature of morphologies and orientations of alloys microstructures is of particular interest because itsets the segregation in the interdendritic region, it governs the inhomogeneous of properties, and itdetermines the formation and distribution of other p
8、hases. Metallography enabled materials scientist tointerpret properties of microstructures. In most cases, 2D metallography has can be successful to revealthe nature of microstructures, however, microstructures of crystalline materials are typically 3D. Theimportance of the full 3D characterization
9、of microstructures is being realized because, in several cases,3D results have revealed that conclusions based on two-dimensional characterization are insufficient oreven incorrect. Rapid advances in theoretical and phase-field modeling technique as well as newexperiments have led to major progress
10、in solidification science, and have improved our fundamentalunderstanding of dendritic growth during the past many years. Phase-field models PFMs wereproposed 20 years ago to tackle the difficult problem of crystal growth. From the original work byCollins and Levine 5 for the simulation of diffusion
11、-limited crystal growth, the PFMs approach hasbeen extended to dendritic growth by Kobayashi 6, and later by Karma and Rappel 7-10, Wheeler et al.11,12and to dendritic microstructures coarsening or Ostwald ripening by Mendoza 13, stress-induced4045instabilities in solids by Kassner 14 and solid stat
12、e transformation by Militzer 15 etc. Recently, phasefield method becomes increasingly popular in analysis of complex microstructures in 3D because of itsadvantages in the description of microstructures morphologies evolutin and orientations ofpolycrystalline materials.However, most of previous phase
13、 field simulation of microstructure focused on metals withface-centered cubic fcc or body-centered cubic bcc phase, such as Al, Cu and Ni, and metals withFoundations: Doctoral Fund of Ministry of Education of China, No. 20090002110031 Brief author introduction:WANG Mingyue 1983- , Male, Ph.D., Three
14、-dimensional Characterization ofMaterials: Microstructures and Mechanical BehaviourCorrespondance author: JING Tao 1965- , Male, Professor, Solidification & CAD/CAE. E-mail:jingtao-1-hexagonal close-packed hcp phase structure, such as Mg, Zn and Cd etc. were rarely referred to. Aspecific feature of
15、Mg alloys is the hexagonal structure of the crystal lattice. In fcc and bcc materials,the001? direction is found to be the direction of the dendrite stem. In hcp materials, different stemdirections have been reported for different materials. In Zn and Cd, it is well documented that10 1 0? is505560th
16、e stem direction 16-17. In other hcp materials, however, other stem directions have also been reported.In magnesium and magnesium alloys, there are only a few investigations reported in the literature, andthe results are divergent. In a previous investigation by Pettersen and Ryum 18,19, it was foun
17、d that thedendrite stem direction during unidirectional solidification of the alloy AZ91 was11 2 0?. They alsofound that the dendrite arms were lying in the basal plane and in the 1 1 01 planes of the11 2 0? zone.For hcp-phase, pure zincs crystal structure is similar to magnesium, the equilibrium sh
18、ape of the solidin its melt was also measured 20, showing a lens shaped structure. This indicates that instabilities in thebasal plane might be favored also in the growth structure. In the context of phase field modeling ofdendritic solidification with none-cubic symmetry, the work of Semoroz et al.
19、 21 deals with the growthof zinc dendrites in coatings of steel. Also Karma presented a phase field study of the sensitivity ofpreferred growth directions on the anisotropy function of the surface energy in 3D 22. Experimentalresults indicate that the dendritic morphology of metals with hcp structur
20、e is totally different frommetals with fcc or bcc structure because of the different crystal lattice. Fig. 1 shows the typicalfcc-phase and hcp-phase dendritic microstructures morphologies in aluminum alloys and magnesiumalloys, respectively.65Fig. 1 Dendritic microstructures morphologies of fcc-pha
21、se obtained by optical microscope left block andhcp-phase by SEM right block in typical Al alloys Al?wt.7%Si and Mg alloys AZ91D , respectively.It should be noted that the properties of microstructure of most technical alloys are typically7075polycrystalline and 3D. Obviously, the modeling of polycr
22、ystalline solidification processes is also achallenging work because of the crystallographic orientations of all grain-to-grain need to beincorporated, and we have presented a new single variable phase field algorithm to addresspolycrystalline nucleation and growth, and papers about that will be see
23、n in other journals. The clarityof 3D dendrites morphologies of Mg-based alloys microstructures is the main aim of this paper. Herein,the research is focused on the free dendritic solidification growth of single crystal grain.1 Phase-field model1.1The Phase-field equationsThe total free energy of th
24、e two-phase system is described by a phenomenologicalGinzburg-Landau model as80?V? 2n?22? 2n?2?cdV 1 where F is the total free energy of the system, f ,u,c is the free energy density function, is the phasefield variable parameter ranging from negative one in the liquid to positive one in the solid,
25、u and c are-2-the temperature and concentration, respectively, and n and n are the gradient energy andconcentration field coefficient, respectively. The governing equations based on thermodynamic theory85for the phase-field coupled with temperature field and solute field can then be expressed as fol
26、lowsn?t?Fc,?FT,? DT2T L?h?t 2 where n is the kinetic coefficient, MC is a concentration mobility parameter and is setasDS hDL DS RT VM , h is the solid fraction given by 23 2 , DT is the thermal diffusioncoefficient, L is the latent heat.901.2Thermodynamics descriptions of alloy systemThe system Gib
27、bs free energy description based on ideal solution approximation and regularsolution approximation are not suitable for simulation of Mg?Al alloy system regarding the complexthermodynamic relationship. Instead, using Redlich-Kister binary excess model of the extendedsubstitutional-regular-solution a
28、pproximation by the CALPHAD database seems to be most close to the95essence of technology. The study of microstructures numerical simulation by phase field method inmulticomponent alloys coupled with CALPHAD can be found in Refs. 23-28. In this paper, the freeenergy density function of the system co
29、nsists of the free energy of the bulk phases and an imposedparabolic potential which is given and can be written ashcp L 3 100where g 212 , WA and WB are the height of the parabolic potential which will be determinedas W A3? A2TMA? A,W B3? B2TMB B.The definition of the free energy of solution based
30、as the extended substitional-regular-solutionapproximation whose excess free energy is based on the Redlich-Kister binary excess model is shown105as belowLhcpLMgSMgLSVMVM 4 5 ?J mol 1?LL012000? 8.566T , L1L 1894? 3T , LL2 2000, LS0 1950? 2T , L1S? 1480? 2.08T , LS2 3500In a two-phase system, the Gib
31、bs free energy for the liquid phase, hcp-Mg phase and others110thermodynamic parameters obtained from thermodynamic SGTE database 29 by the Pandat softwareare expressed respectively ashcpliquid165.097? 134.839T 26.185T lnT 0.0004858T 21.39367?10?6T 3 78950T1 8.0176?10?20T 7hcp115liquid1.3271.21? 211
32、.207T 38.5844T lnT 0.018532T 2 5.76423?10?6T 3 74092T1 7.9337?10?20T 7The anisotropic functions of interface energy and mobilityAt the center of our understanding of 3D dendrite morphology is the concept of interfacialinstability which result in the anisotropy form of the crystal-melt interface free
33、 energy and kineticcoefficient. In the context of phase field method, the discusses about anisotropic function in 3D is120deficient, however, the anisotropic formula in cubic metals 7,30-32 based on cubic harmonics are-3-accepted by and large within the metallurgical community, and yield the whole 3
34、D cubic symmetrydendrites. So far, the discrepant representations of anisotropy for hexagonal metals are yet put 27,33,34but failed to yield 3D hexagonal dendrite morphology, at the same time, theoretical and/ormathematical demonstrations remain to be deficient. We believed that the anisotropic prop
35、erties in 2D125130135 0001 basal plane is no doubt that six-fold preferred growth directions in the 2D section planedominate the dendritic structure and the angle between the primary and the secondary of dendrite is 600,which also accords with the symmetry of lattice though some atypical growth patt
36、ern can also beobserved. The solidification of Mg alloys melts is the focus of present study, according to our known,compared to pure metals, solidified patterns and dendritic growth directions of alloys melts are morecomplex and various than previously thought, although the two mechanisms are very
37、similar only froma mathematical point of view, the former is largely controlled by the diffusion of heat and the latter iscontrolled by the diffusion of concentration. From the combined point of crystallographic point grouplattice and experimental phenomena, we are more willing to believe that cryst
38、al growth in dihexagonalD6h metal Mg in 3D scale should be seen as the overall stacking growth of 2D which is different fromthe point of other materials scientists or physicists, however, this model can explain many phenomenain experimental observations. Of course, the more complete and powerful stu
39、dy are needed to do.An anisotropic function of interfacial free energy and mobility, which reflects underlyingcrystalline characteristics of hcp lattice, is proposed:n 0 11 n6 15n4n2 15n2n4 n6 2n6 ,n0 11 n6 15n4n2 15n2n4 n62n6? 6 140where ni is unit vector and is given nii2and i represents x, y, z,
40、respectively, 0 and 0are the mean values of interface free energy and kinetic, 1, 2, and 1, 2 are, respectively, the first- andsecond-order anisotropic parameters of crystal-melt interface free energy and mobility. However noreliable experimental data are known for the values of anisotropies of both
41、 the surface energy and thesurface mobility for Mg alloys. To date, few simulations 32,35 and experiments have been undertaken145150155to explore the anisotropies of and in hcp, relatively, than fcc metals. The data of the magnitude andanisotropy of the solid?liquid interface free energy and kinetic
42、 are selected based on the simulationresults of molecular dynamics by Sun et al. 32,35.2 Results and discussionTo yield well-developed 3D dendritic morphologies, phase-field simulations of equiaxed dendriticsolidification growth in undercooled melts were carried out. A nucleus was located in underco
43、oledmelts and equiaxed solidification process, governed by both phase-field equation accompanied withassociated temperature and solute field equations, was carried out. All parameters in the simulation areshown in Table 1.Tab. 1 Thermodynamic data of Mg alloy system and parameters used for simulatio
44、n.DescriptionsMelting point K ParametersTMMgValue922ParametersTMAlValue933.3721DL1.410?5DS3.610?9Latent heat of per volume J?cm?3 The mean of surface energy J?cm?2 The width of S/L interface cm The mean of energy & mobilityThe first-order magnitudeThe second-order magnitudeInitial composition at. &
45、temperature K Thermal diffusion coefficient cm2?s?1 Specific heat capacity J?mol?1?K?1 Molar volume cm3?mol?1 LMgMgMg012C0DTCPVM622.141.1510?56.1110?61.1510?50.060.0070.090.824.614.0LAlAlAl012T01064.61.90210?54.5010?60.180.070.01890-4-Grid size cm x410?7Phase-field calculations, based on two differe
46、nt anisotropic parameters, yield the entire dendriticsolid?liquid interfacial S/L morphologies shown in Fig. 2, corresponding to the pattern of hexagonaldipyramid. It should be noted that anisotropic Eq. 6 may not yield various patterns, which may exist160165170175180185190in alloys microstructures, because they arent extended to more multi-order. Crystal-melt morphologies,however, in hcp metals or alloys e.g. Mg, Zn, Cd, Ti etc. should evoke more attentions within thescientific community, after all, great theoret