Simulated evolution process of core-shell microstructures Simulated evolution process of core-shell microstructures

Simulated evolution process of core-shell microstructures

  • 期刊名字:中国科学G辑(英文版)
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  • 论文作者:QIN Tao,WANG HaiPeng,WEI BingB
  • 作者单位:Department of Applied Physics
  • 更新时间:2020-11-10
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论文简介

Science in China Series G: Physics, Mechanics & Astronomy◎2007 SCIENCE IN CHINA PRESS包Springer .Simulated evolution process of core-shellmicrostructuresQIN Tao, WANG HaiPeng & WEI BingBo*Department of Applied Physics, Northwestern Polytechnical University, Xi'an 710072, ChinaThe evolution process of core-shell microstructures formed in monotectic alloysunder the space environment condition was investigated by the numerical simula-tion method. In order to account for the effect of surface segregation on phaseseparation, Model H was modified by introducing a surface free energy term intothe total free energy of alloy droplet. Three Fe-Cu alloys were taken as simulatedexamples, which usually exhibit metastable phase separation in undercooled andmicrogravity states. It was revealed by the dynamic simulation process that theformation of core-shell microstructures depends mainly on surface segregationand Marangoni convection. The phase separation of FesCu3s alloy starts from adispersed structure and gradually evolves into a triple-layer core-shell micro-structure. Similarly, FesoCuso alloy experiences a structural evolution process of“bicontinuous phase→quadruple-layer core-shell→triple-layer core-shell", whilethe microstructures of FegsCu6s alloy transfer from the dispersed structure into thefinal double-layer core-shell morphology. The Cu-rich phase always forms the outerlayer because of surface segregation, whereas the internal microstructural evolu-tion is controlled mainly by the Marangoni convection resulting from the tempera-ture gradient.space environment, Fe-Cu alloy, phase separation, core-shell structure, numerical simulationLiquid phase separation of monotectic alloys has aroused great interest in the fields of materialsphysics and space sciencel- '. The earlier researches show that surface segregation and Marangoniconvection resulting from the temperature gradient play an important role in the microstructuralevolution under the space environment conditionl24. Due to the opacity of metals and the formrequirements of the experiments, it is quite difficult to carry out theoretical investigation deeply.Therefore, the evolution process and the micro-mechanism are still poorly understood. Numericalsimulation has become an important method to explore phase separation. However, it has beenexclusively implemented to study the pattern formation of polymers and little attention has beenReceived February II, 2007; accepted May 29, 2007doi: 10. 10143300700457tCorresponding author (email: bbwei@nwpu.cdu.cn)Supported by the National Natural Science Foundation of China (Grant Nos. 50121101 and 50395105) and the Doctorate Foundation of North-westerm Polytechnical University (Grant No. CX200420)www.scichina.com www.springerlink.comSci China-Phys Mech Astron l Aug.2007 lvol. 50 I no. 4 | 546-552中国煤化工YHCNMH G.paid to the metal systemsb.o. Furthermore, the previous numerical studies are primarily focused onthe phenomena in rectangular geometries at a fixed temperature, and lttle work deals with theprocesses in other geometries with nonuniform additional fields. Under the space environmentcondition, liquid droplets form perfect spheres with the temperature gradient inside"", therefore,the traditional simulation methods are inapplicable.Model H is of great success in the simulation of phase separation8. It includes the coupling ofthe flow field and concentration field, and is frequently used to study the bulk phase separation. Itsapplication in the phase separation with surface segregation is still limited.Fe-Cu peritectic alloy is one of the most important metastable monotectic systems. Its equilib-rium phase diagram is shown in Figure 1. The matrix decomposes to Fe-rich and Cu-rich phaseswhen it is undercooled to the miscible zone over a broad composition range. Due to the thermo-dynamic metastability, the phase separation can easily be suspended by the growth of yFe dendrites,and produce various microstructures' . Although some researches have been carried out to ex-plore its metastable phase separation behaviors, further efforts are still needed to reveal the mi-crostructural evolution process and the mechanisms.In this paper, numerical simulation is performed to investigate the microstructural evolutions ofhighly undercooled Fe-Cu alloys droplets under the space environment condition and the effects ofthe surface segregation and temperature gradient are analyzed. Three selected compositions in thesimulations are shown in Figure 1.18008FFesCus Fe。Cuso Fe,Cus600| yrF1500 |yFe+L14000406(80100FcCCu (at%)1 Mathematical model and numerical methodThe liquid phase separation can be described by Model H, given in the dimensionless form aslo.t0∞=-()+V.0l-0ovO5+v.ζ1)where φ denotes the molar fraction of the segregating component, i.e. Cu in this work, T is the time,v the local velocity, δ the Lagrange variation operator, and ξ the random Gaussian white noise withthe fluctuation amplitude ε, and F the free energy functional.Previously, only the bulk free energy Fr and the contribution from the concentration gradientQIN Tao et al. Sci China Ser G Phys Mech Astron I Aug.2007 Ivol. 50 I no. 4」546-552547中国煤化工MYHCNMH G.Fgrad were considered when describing the system free energy F. In fact, the surface free energy Fsalso plays a crucial role in phase separation. In the presence of surface segregation, the surface freeenergy Fs has to be involved. Thus, F should be written byF=F+Fgrad+Fs.(2)It is reasonable to take Fb to be one of the regular solutions, i.e.F=Fo + θ[φlnφ+(1- φ)ln(1- )]+ 2Q(l -中),3)where Fbo is the constant part, θ the reduced temperature, defined as T/TL, and TL the critical tem-perature. Fgrad is given as .Fgma=号V2φ.(4)Fs is taken asl2])Fg=Fso-H% +士g3,5)where φs represents the surface concentration, and Fso, H and g are constants, describing the sur-face preferential attraction properties.In small droplets, the Reynolds number is less than the magnitude of 10",therefore, the localvelocity can be taken proportionally to the local body forcef=-VF [15]. Expending VF gives8Fv=-a.pV6)φwithρRTI(7)6πD_ Mη'where ρ is the mass density, l the interface length!", Rg the gas constant, DL the diffusion coeffi-cient, M the molar weight, and η the viscosity. a reflects the fluidity of the alloy melt, and the largerit is, the more severely the flow field responds to the local force field.Eqs. (1)- -(7) are supplemented with the following boundary conditions at the surface R =Ro,n.V则p =0,8)n.voF=0,9)8φ|where n gives the normal direction at the surface. The incompressibility condition on the fluid(V .v=0) is not applied here, and the justification is that this effect is not significant as numeri-cally validated by Chen and ChakrabartilsTo incorporate the influence of the temperature gradient on the phase separation, a nonuniformtemperature field is applied. It decreases linearly along the radial distance from the center to thesurface with the total temperature difference of5 K. The temperature at the surface is set to be0.8TL, corresponding to a highly undercooled state.The initial concentration is set at φo (= 0.35, 0.50 and 0.65) with random fluctuations of am-plitude 0.05. The initial local velocity is zero. The concentration fluctuation amplitude ε in thesimulation is 0.01 and no fluctuation of the velocity field is considered. The surface parameters areH= 0.35 and g=0.4. ε, H and g are determined empirically. a is estimated as 1000 for the alloys.The above equations are solved in two-dimensional Cartesian coordinates using an explicit finite548QIN Tao et al. Sci China Ser G-Phys Mech Astron I Aug.2007 Ivol. 50 I no.4. | 546-552中国煤化工MYHCNMH G.difference scheme. The grid spacings are Ax= Oy= 1 with a fixed lattice size Ro= 100, and the timestep is △r= 0.0001 to ensure the stability. The calculations were performed in a Lenovo 1800cluster system.2 Results and discussion2.1 Phase separation of FeasCu35 alloyThe microstructural evolution snapshots from the simulation of FesCu3s at different times areshown in Figure 2. Obviously, in the early stage, surface segregation occurs prior to the bulk de-composition, and leads to the formation of a Cu-rich layer followed by a Fe-rich layer at the surface(t= 0.1). In the bulk, as the minor phase, the Cu-rich spheres form and grow diffusively (t= 5). .After τ= 5, the hydrodynamic effect begins to behave visibly. The flow field quickly responses tothe local force field and the coalition becomes much faster, and the Cu-rich spheres grow rapidly(τ= 50). Since the thermo-capillary effect is inherently included in the numerical model, the re-sultant Marangoni motion of the Cu-rich spheres towards the hotter, i.c. the central area, can beexplicitly observed in the simulation (τ= 50, 100, 150 and 260). The spheres aggregate at thecentral part and the triple-layer core-shell structure forms in the end.In order to explore the influences of surface segregation and Marangoni convection on the con-centration field during phase separation, the radial Cu distributions at different times werecomputed, as presented in Figure 3. The angular Cu concentration Pav denotes the average valueover the range of[r- 0.5, r+ 0.5] and every point set is averaged over 10 independent runs. Ac-cording to Figure 3, due to the surface enrichment of Cu, the concentration profile forms a crest anda trough near the surface in the beginning (t= 0.1). As the phase separation continues, in theFe-rich layer, more Cu is exuded into the two lateral sides and more Fe is absorbed from them.Consequently, new layers rich in Cu and Fe form at the inner side. Therefore, the concentrationprofiles are shaped as waves by this chain reaction (t= 5). The Cu-rich phases gradually coarsenand move towards the center area, and consequently, the waves propagate deeper into the bulk (t=150).-.-. τ=150--- t 260+ CenterSurfaccτ=0.1τ=5τ=50、 。.2 t.00.0.8τ= 100τ= 150.τ= 260 .r/R,Figure 2 Snapshots from microstructural evolution proc-Figure 3 Cu concentration Aov versus radial distance.ess ofFesCus alloy. The grey level varies linearly betweenblack and white, corresponding to the equilibrium concen-trations 4≈0.86 and x2≈0.15, respectively.QIN Tao et al. Sci China Ser G Phys Mech Astron I Aug.2007 Ivol. 50 I no. 4」546-552549中国煤化工MYHCNMH G.Based on the above investigation on the concentration distribution, the thickness and the growthrate of the surface Cu-rich layer were determined to further study the growth behaviors of thesurface layer. The results are exhibited in Figure 4. It can be seen that the surface layer growsrapidly at first, gradually slows down and reaches a plateau. This deeleration can be explained bythe competition between the surface segregation of Cu atoms and the Marangoni convection of theCu-rich spheres. In the beginning, it is very easy for Cu to diffuse to the surface, and the growthspeed is quite large. As the phase separation progresses, the Cu-rich spheres gradually move to thecentral part, less and less Cu diffuses through the Fe-rich layer to the surface layer, and conse-quently, the thickening slows down.In order to understand the thermodynamics at different stages of phase separation, the evolutionsof the total and partial free energies were also calculated in the simulation, as ilustrated in Figure 5.The constant parts Fr and Fs are set to be zero when computed according to eqs. (3) and (4), re-spectively. All the data sets are averaged over 10 independent runs. It can be seen that the total freeenergy F decreases monotonously with the time and the surface energy Fs quickly drops to theminimum value and keeps at a rough constant. The bulk free energy Fb and the gradient energyFgrad evolve differently. At the early time, the matrix decomposes and interfaces quickly form.Consequently, Fr decreases and Fgrad increases steeply. During this time, the structural evolution isdictated by Fo. When t exceeds 2, Fgad begins to drop, and the reduction of the interfacial energybecomes dominant. This effect is embodied in two aspects: one is the coarsening of the phasedomains to reduce the interfacial area; the other is the Marangoni motion towards the hotter area tominish the interfacial energy intensity.12200 t10,260.1习dL./dr-4500-! 0.01- 500050 100 130 200 250 0.001-500010- 1Figure 4 Surface layer versus dimensionless time.Figure 5uriation of free energy with dimensionless time.2.2 Phase separation of FesoCuso alloyThe microstructural evolution process of FesoCuso alloy is illustrated in Figure 6, and the corre-sponding concentration profles are shown in Figure 7. At this composition, the surface segregationprocess is similar to that of Fe6sCu3s. The surface layer is formed and it thickens quite fast initiallyand then grows slower and slower to the maximum value. Inside the droplet, the matrix separates tobicontinual phases instead of spheres, due to the same amount of the two components. After thehydrodynamic effects induced by the interfacial energy begin to dominate, the microstructureevolves from bicontinual phases to a quadruple-layer core-shell, and to a triple-layer core-shelleventually. The growth behavior of the surface layer and the free energy variation are similar tothose of Fe6sCu35 alloy as shown in Figures 4 and 5.550QIN Tao et al. Sci China Ser G-Phys Mech Astron I Aug.2007 Ivol. 50 I no.4. | 546-552中国煤化工MYHCNMH G.Surface -f-Center i).6上τ=0.1τ=Iτ= 200.4-.-. τ= 50.2 t0.0).2.40.60.81.0τ= 30τ=50τ= 200rlR。Figure6 Simulated microstructural evolution for FesyCusoFigure 7 Cu concentration versus radial distance forFesCuso aloy.2.3 Phase separation of Fe3sCu65 alloyFigure 8 gives the microstructural evolution of Fe3sCu6s alloy, and the corresponding concentrationprofle variations are plotted in Figure 9. Since the segregating component is the majority, noFe-rich layer forms near the surface, whereas the concentration fluctuation similar to those in theformer alloys is still aroused, as can be seen from Figure 9. The minor Fe-rich phases turm out inspheres, and grow through diffusion at the early stage and coagulate by Marangoni convection atlater time. The microstructure evolves from a dispersed morphology to a double-layer core- -shell.0.8 F).6 Kτ 0.1τ= Ir= 10- Centersurface - -).4 Fr=01--- τ=10).2 上t= 50τ= 250τ= 400).8alloy.Finally, it is mentioned that the present approach can be extended to deal with the phase sepa-ration during rapid solidification by adding the effect of the temperature evolution, and the simu-lations of other immiscible alloys can also be applied by choosing proper free energy functionalsand parameters. Owing to lack of experimental results describing the time evolution of the mi-crostructures of those alloys, a comparison of the present simulations with other investigations islimited. It is expected that the above results will provoke fresh experiments on these problems.QIN Tao et al. Sci China Ser G Phys Mech Astron I Aug.2007 Ivol. 50 I no. 4」546-552551中国煤化工MYHCNMH G.3 Conclusions(1) Model H has been modified by taking into account the effect of surface free energy so that itbecomes applicable to the liquid phase separation process under the influences of suface segre-gation. As a practical application, the evolution process of core-shell microstructures of under-cooled Fe-Cu alloys in space environment was numerically simulated in the light of this modifiedModel H.(2) According to the simulated evolution process, the phase separation of FesCu3s alloy ex-periences a transformation from a dispersed structure into a triple-layer core-shell microstructure.FesoCuso alloy follows the structural evolution route of“bicontinuous phase - > quadruple-layercore-shell - →triple-layer core-shell". Similarly, Fe3sCu6s alloy derives its final double-layercore-shell microstructure from an initial dispersed morphology.(3) At the beginning of phase separation, the evolution of structural morphology is controlled bythe reduction of surface free energy and bulk free energy. The effect of surface segregation resultsin the location of Cu-rich phase at droplet surface and induces the appearance of concentrationfluctuation. Later on, the interface free energy becomes the dominant factor. The Marangoniconvection caused by the temperature gradient broadens the concentration fluctuation, makes itspread toward the droplet center and controls the final evolution of internal microstructures.The anuthors are graleful to Luo Bingchi and Chang Jian for helpful discussions.Carlberg T, Fredriksson H. The inflence of microgravity on the solidification of Zn Bi immiscible aloys. Metall Trans, 1980,12: 669-6752 Wang C P, Liu X J, Ohnuma I, et al. Formation of immiscible alloy powders with egg-type microstructure. Science, 2002,297: 990-9933 Tegze G, Pusztai T, Grainasy L Phase field simulation of liquid phase separation with fluid flow. Mater Sci Eng A, 2005,413-414:418-422LuX Y, Cao C D, Wei B. Microstructure evolution of undercooled iron-copper hypoperitectic alloy. Mat Sci Eng A, 2001,313: 198- 2065 Tanaka H. Interplay between wetting and phase separation in binary fluid mixtures: Roles of hydrodynamics. J Phys:Condens Matter, 2001, 13: 4637- 4674Huo Y L, Zhang H D, Yang Y L. The morphology and dynamics of the viscoelastic microphase separation of diblock co-polymers. Macromolecules, 2003, 36: 5383 - 53917 Liu X R, LuX Y, Wei B B. Rapid monotectic solidification under free fall condition. Sci China Ser E- Eng Mater Sci, 2004,47(4): 409 - 420Hohenberg P C, Halperin B I. Theory of dynamic critical phenomena. Rev Mod Phys, 1977, 49: 435- 4799 Nakagana Y. Liquid immiscibility in copper- iron and copper-cobalt systems in the surpercooled state. Acta Metall, 1958,611): 704-710) LuX Y, Cao C D, Wei B. Metastable phase separation of FesoCuso hyperitectie alloy under space simulation conditions.Acta Metall Sinica, 1999, 12(2): 198 - -20411 Vladimirova N, Malagoli A, Mauri R. Two-dimensional model of phase segregation in liquid binary mixtures. Phys Rev E,1999, 60: 6968 - - 697712 Feng D, Jin G J. Condensed Matter Physics(D(in Chinese). Beiing: Higher Education Press, 2003. 610- 61413 Mautits N M, Zvelindovsky A V, Sevink G J, et al. Hydrodynamic efects in three-dimensional microphase separation ofblock copolymers: Dynamic man-field density funtional approach. J Chem Phys, 1998, 108: 9150- 91544 Mauri R, Shinnar R, Triantafyllou G Spinodal decomposition in binary mixtures. Phys Rev E, 1996, 53: 2613- 262315 Chen H, Chakrabarti A. Surface -directed spinodal decomposition: Hydrodynamic effects. Phys Rev E, 1997, 55: .5680- 5688552QIN Tao et al. Sci China Ser G-Phys Mech Astron I Aug.2007 Ivol. 50 I no.4. | 546-552中国煤化工MYHCNMH G.

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