Modeling of Microstructure Evolution during Directional Solidification Process Modeling of Microstructure Evolution during Directional Solidification Process

Modeling of Microstructure Evolution during Directional Solidification Process

  • 期刊名字:武汉理工大学学报(材料科学版)英
  • 文件大小:522kb
  • 论文作者:LI Qiang,LI Dianzhong,QIAN Bai
  • 作者单位:School of Material Science and Engineering,Institute of Metal Research
  • 更新时间:2020-11-10
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论文简介

Vol.20 No.2Journal of W uhan University of Technology - Mater. Sci. Ed.Jun. 2005Modeling of Microstructure Evolution during DirectionalSolidification ProcessLI Qiang' 2 LI Dianzhong QIAN Bainian2( 1. School of Material Science and Engineering , Shengyang University of Technology , Shenyang 110023 , China ;2. Institute of Metal Research , Chinese Academy of Sciences Shenyang 1 10016 China )Abstract : A model was presented to dlescribe the microstructure evolution during the directional solidificao-tion process. In this model , the problem of diferent properties in the solid and liquid phuase uwas solred by makingthe properies continuous al the solid liquid inteface. Furthermore ,a random noise rwas incorporated to reflethe anisotropic grouth. Moreoter ,the weraging solute conservation uwas dereloped to heep the total solute conser-vation in the interface region. A simple ingot was simulated by this method , the model can represent the mi-crostructure evolution , solute concentration redistibution , micro- segregation and the columnar-to- equiaxed transi-ion.Key words : directional solidifiction ; microstructure evolution ; columnar to- equiaxed transition ;micro-segregation1 Introductionterface was continuous in thermal solute. In fact ,thethermal physical difference in both solid and liquid phaseswill lead to a heat transferring boundary different from theDirectional solidification is an important materialformer model. In addition , beside the solid/liquid inter-processing method to obtain the specifie properties , whichface, there is a solute concentration gap to reflect theis used widely in aerospace , super alloy etc. Because thechemical potential equilibrium. If the solute concentrationdirectional solidification can get a strong preferential ori-is continuous beside the interface , the chemical potentialentation microstructure that has superior properties in thewill be in unequillibrium.preferential orientation than those in other orientations ,In order to overcome these problems , a continuouslots of researchers used the phase field method to simulatemodel is presented. Firstly , the whole simulated zone is .the solidification structure 13]. Karma and Rappel useddivided into three possible zones , solid , liquid and inter-this method to simulate the dendrite evolution in three di-face , in which the physical properties in different zonesmensions 61. However , a phase function has to be intro-are different except for the density , and then in the inter-duced into the phase field model that had no real physicalfacial region the physical properties are equal to the summeaning. Furthermore ,the phase field required theof solid fraction multiplying the solid physical propertiesmeshed size be smaller than the thickness of interface lay-and liquid fraction multiplying the liquid physical proper-er , which limits the calculating scale. In addition , theties, which make the discontinuous physical propertiesthickness of interface is variable , which not consists withcontinuous. In addition , in this model a random noise isreal solidification structure.incorporated into the model to reflect the anisotropicThe front-tracking methoct4 5 J can track the precisegrowth. Moreover ,because the heat transfer model isposition of interface. But this method needs to know theadopted as a macro-heat transfer model,the model can bedetail of interface , and most of calculating times are con-extended to the macro-scale simulation .sumed in iteration of solving interface position ,whichlimits the calculating scale.2 Mathematical ModelCellular automaton method671 has a simple transi-tion rule and physical background of solidification. C FThe model is based on the following assumptions :Hong used the KGT model and iteration method to obtainthe density of material is not changed during the phasethe velocity of dendrite tip , then by using this velocity tctransition process ; the solute can not diffuse across thecalculate the growth length and solid fractiort 81. However ,sol中国煤化工the solute in the liquidphasFunid phase ,and the solutemost of the researchers assumed that the thermal physicsin IYHCN M H G.e in the solid phase ;thein both solid and liquid phases was the same , and the in-influence of liquid flow is neglected ; in the whole simu-lated zone there are only three kinds of states , liquid( Received :0ct. 212004 ;Accepted Jan. 18 2005 )phase,solid phase and interface. The liquid phase stateLI Qianf李强) E-mail :qli@ imr. ac . cn* Funded 芳芳数据”Program of Chind No. 2001AA339030 )is taken as zero , and the state of interface zones is asumed as a random number bigger than zero , whose stateVol.20 No.2L Qiang et al :Modeling of Microstructure Evolution during Diretional....95number is not changed but the solid fraction.interface , subscript s and l stand for the solid and liquidIn the solid phase zone , energy conservation equa-phase ,respectively. h the heat transfer coefficient ( W/tion is :m2K ), q the heat flux through the boundary.pCpataTv.( K。V T)(1)The above Eq.( 7 ) can be discretized as equations( 13)( 14):species conservation equation is :V.= dI-kF(1--DC(i-1 j))(i-1 j)+ .at= V( D, VC、)(2)(1-C(i+1门C*(i ;i)In the liquid phase zone , energy conservation equa-C*(i i)')f(i+ 1 j)]+k:pD。C(i-1j))f(i-1 1)+pCpi=v(K1VT)(3)d(1- kpkp° C*(i j).C(i+1门r Cc(n)6i+1j)] (13)a7= v(D VC). (4)V, =d(1- 1D、f(1-C(i j-1))fi j- 1)+In the solid-liquid interface zone , energy conserva-C"(i j)tion equation is :C(igj+ 1)?)ri j+ 1)]+af。pCp= V.(KV T)+pLg(5)kpD。[(1 -C( i j-1))f(i j-1)+Cp=Cp f.+Cpi fi k=h: f.+hi fi f.+fi=1(1- hpkp" C*(i ;)(6)C(ij+l)In the flux of the solute across the solid/ liquid inter-(1 -C((;)rij+1)]( 14)faceif11,12]:where , C* is the interface solute concentration , which is(1- kp)C* V=D(Cλ-D(2) (7)the liquid solute concentration in the liquid side of inter-anface ,a the meshed size.The solute concentration partition ahead of solid-liq-Based on the KGT model it can be obtained that theuid interface is :solute concentration in solid-liquid interface zone is givenC、=h邸C*a511] :boundary conditions :aC(9)T- Ten+ThK(φ 0)C* = Co+(( 15)non the bottom side of ingot :q=h( Tr- T) (10)the other sides of ingot are adiabatic boundary :where , m is the liquid slope ,h the averaged curvature ofq=0.( 11)solid-liquid interface ,C* the interface solute concentra-initial conditions :tion , Teq the melting temperature of material,T can beT(r=0x y)=T{x y) Qt=0x y)=C{x y)obtained from Eqs.( 1 )63)65). f( φ θ )can be given( 12)as following11].where, ρ is the density ( kg/m3 ),C。the specific heat(φ 0)=1+δ co(4(φ-θ))( 16)(kJ(kg K)). In the solid phase , Cp is equal to Cps ,where ,φ is the normal direction of interface ,0 the pref-and in the liquid phase Cp is equal to Cp , where , in the .erential orientation δ the constant δ = 0.04.Fig. 1 shows that as the solute concentration accumu-solid-liquid interface zone , Cp is equal to the solid frac-tion multiplied by Cp and the liquid fraction multipliedlates from Co to C* in the interface zone , the meltingby the Cp. k the conductivity( W/mt K ), in the solidtemperature will decrease from T to T*。From thisphase k is equam to h。,and in the liquid phase hi is e-point ,it can be seen that if the solute concentration accu-qual to ki ,while in the solid-liquid interface zone ,k ismulates ahead of the interface , the liquidus of metal willequal to the sum of the solid fraction multiplied by k。 anddecrease. If T* < T , there is no possibility for the cellthe liquid fraction multiplied by the ky. Ln is the latentto nucleate or be captured .The surface tension also plays an important role inheat released by phase transition from liquid to solid( kJ/kg),f。the solid fraction. The last term in the right sidedetermining the morphology stability .of Eq.( 5) only exists in the solid-liquid interface zone.中国煤化工Zf(h)T is the temperature( K ), T the surround temperature ,YHCNMH GN+1厂)( 17)t the time( s ) and C the solute concentration( % ). Inwhere , N is the total number of neighboring cell. In thisthe solid phase and liquid phase D the selected as D。anemodel we adopted the three nearest layers of cells to cal-D( m?/s ) ,respectively. C * the solute concentration atculate the average curvature. f( h )is the solid fraction ofthe interface. VN the normal velocity of solid-liquid inter-the k:th neighboring cell ,and this is also an averageface , khp the有称据partition coefficient in the solid-liquidmethod to calculate each interface cell curvature .96Joumal of Wuhan University of Technology - Mater. Sci. Ed.Jun.2005Adiabatiewhere , N is the total solid number of the nearest neigh-boring cells ,and M the total liquid number of the nearestneighboring cells. If in the neighboring zone there is an-other interface cell , the interface cell belongs to the N aswell as the M. Cn is the average solute concentration at1|present step ,Cn-1 is the interface solute concentration athe prior step. C。( i ) is the solute concentration at theprior step. C(j ) is the solute concentration in the ithliquid phase ,C( j ) is the solute concentration in the jthCoCliquid phase ,C( i )and C( j ) are all solute concentra-Fig.1 Solute concentration distributionFig.2 The shape oftion at the prior time step. dt is the time step , Cs and C1during solidificationsimulate zonethe solute concentration of the solid part and liquid part inIn the solid-liquid interface zone , the solid fractionthe interface cell. In this model ,the time step is selectedin each time step can be given as 11] :as a variable automatically ,which is dependent on themeshed size and physical properties of material. In thisδf。=(2x + v,- 2x' apaper the time step is :f"=f-1+of。.(18)δt=Min(号(D、_v))(23)where ,of。is the increase of solid fraction in the solid-where , Vmax is the maximum velocity at that step ,a theliquid interface zone f: the solid fraction at the n th timestep , andf-1 the solid fraction at the( n - 1 )th timemeshed size simulated z2one , Min a function that selects astep. In order to rllect the effect of perturbance in frontminimum value amongandD。'D\of the solid/ liquid interface on the morphology ,a randomWhen the liquid temperature is below the liquidus ,noise is generated randomly and incorporated into the sol-there is a possibility for each undercooling liquid cell toid fraction temn , which can be expressed as 121:nucleate. The nucleation model is similar to the modelof。=(1+1-2e)H(ys+y,-;告台adopted by Nastat 1]:aN( t )=Ni=N( 19)2n,( orpTiu a2awhere , η is a constant equal to 0.1 ,ε the small randomaN( 1 )=Ni=Nnumber that ranges from zero to one , whose value is gen-aTia7。0-2u(OT)1aerated randomly in each time step .Because the solute concentration at the interface is .dN= dNe + dN。( 24)much higher than that in the solid and liquid phase , thiswhere μN is the nucleation cofficient in the bulk liq-part of solute will diffuse to the neighboring solid and liq-uid ,pN the nucleation coefficient at the heat extractuid phase , which will decrease the interface solute con-boundary ,N.( t ) the nucleation rate in the bulk liquid ,centration. In order to keep the solute conservative in the .solid/liquid interface zone ,an average interface soluteN( t )on the botom side of ingot. The mucleation possi-bility is proportional to the undercooling and the square ofconcentration is developed.the unit. In this paper ,the μN and μN are 1 x 10( 1/The average solute concentration is given as :mK2 )and 5x 10°( 1/m2K2 )] ,respectively.C= f:. C。+fr Ci(20 )The nucleation possibility is equal to :where , C is the average solute concentration in the inter-dpe := dNdp。=dNc(25 )face region ,C。the solute concentration in the solid-side ofd/interface , and C1 the solute concentration in the liquid-If the liquid temperature is below the liquidus , forside of interface. C, and C1 can be given as following :each cell it will generate a random number that ranges be-C。= hp' Ctween zero and one , if the random number is larger thanC1 =( C-f: C,)fi(21 )the nucleation possbility ,the liquid cell will turm into theinterface cell otherwise it will preserve its state. Once theC* can be obtained from Eq.( 15 ). Due to the so-lute diffusion in both sides of interface ,the average solutecell中国煤化工- ential growth orientation .TheEin is selected as a randomconcentration of interface zone in each time step is in-funcMHCNMH Go to 45. If the pelerer-duced :tial growth orientation is consistent with the maximum un-D。(C。-C(i))Cn = Cn-1- dt(:之(0.5 + 0.5x f( i)x adercooling direction , the grain can growth more quickly .In this paper ,the growth angle is the normnal velocityD. ( C- C(j))(22)of solid-liquid interface ,and the normal velocity angle is.5 +0.5xf(j)xa)given as :Vol.20 No.2L Qiang et al :Modeling of Microstructure Evolution during Diretional....97φ= arctan( V,/V,)(26)gy at different calculating steps . In Fig.3 , the black colorWhen the interface is completely solidified , it willstands for the liquid phase zone pand the other colors standcapture the neighboring liquid cell to become the interfacefor the non-liquid zone. Fig. 3( a ) shows the microstruc-cell and change the neighboring phase state number as itsture morphology at the beginning of solicitation in whichphase state number. The capturing possibility is given as .the fine equiaxed grains form along the heat extractfollowing :boundary since the temperature at the boundary is firstlyP。= cosφ.(27)cooled below that of the liquidus. As the solidificationwhere , P。is the capturing possibility ,φ the nornal angletime increases ,the columnar grain forms and its growth di-rection is perpendicular to the heat-extracted boundary.of interface velocity .At the beginning of solidifcation , there is only aBecause the effect of velocity in the dendrite tip is theliquid phase. As the solidification time increases ,themaximum and in this part of undercooling is much largerheat is extracted from the bottom side of ingot , whichthan those of other parts ,the tip of columnar is much thin-causes the liquid temperature descend. When the liquidner than its stem ,which can be seen in Figs.3( b)( f).termperature is below that of the liqudus ,there is a possi-During the columnar grain growth process .columnar grainsbility for the underooling liquid to nucleate. If the liquidcompete with each other that make some columnar graincell is nucleated , its state number is changed to a randomdead" and others grow bigger and bigger ,as shown ininteger number that is bigger than zero and becomes theFigs.3( c)( h).In Figs.3( g)(h), it is also shown thatinterface cell. When the solid fraction of interface cell isthe equiaxed grains form ahead of the columnar , whichequal to one ,it is assumed that the interface is completelylimits the growth of the columnar grain ,that is so-calledtransormed into the solid phase. Once the solid cellcolumnar-to- equiaxed transitior( CET ). In addition,com-forms , it will capture the neighboring undercooling liquidpared with the temperature gradient ,the solute gradient iscell to become interface cell. The iteration of the aboveso small that the interface of columnar is stable. There-steps continues until it reaches the total calculating steps.fore ,the second dendritic arm can not appear at the inter-3 Modeling Parameters and Boundaryface of columnar grains. From Fig.3 ,it can be seen thatthe single heat extract boundary can get well orientationConditionscolumnar grains during the solidification process . .Fig.4 shows the solute distribution during the solidi-In this model , the material for simulation is selectedas Fe-0. 6% binary alloy , whose physical properties arefication process. The right side of the figure is color barslisted in Table 1.that stand for the solute concentration.In the Fig.4 , theTable 1 The simulation parametersred color stands for the solute concentration near 0.3%,ParametersValuewith the red color being weakened gradually , it means theDensity/( kg/m2’)7300solute concentration increases gradually. In Fig. 4( a ),the columnar grains grains form on the bottom side of in-Themal conductivity of solid phase ,W/(m? K)Themal conductivity of liquid phase ,W/( m2 K )got , which is red color ,and its solute concentration isSpecific heat of solid phase kJ/(kg K)650nearly 0. 4%, and among the columnar grains the soluteSpecifie heat of liquid phase kJ(kg K)800concentration is above 0. 6%. As the solid time increas-Solute difusivity in solid phase m'/s5e*I0s ,the columnar grains grow longer and longer ,on theSolute difusivity in liquid phase m2/s2e-’bottom side of ingot the difference of solute concentrationsLatent heat kJ/kg284among the columnar grain decreases gradually , which canLiquid slope ,K/%-82be seen in Figs.4(b)(c)(d). In Fig. 4( d) the soluteThomson-Gibbs cofficient K mconcentration difference at the root of columnar grains isPartition cofficient0.6not very distinct. Meanwhile ,on the upper side of colum-Melting temperature K1743nar grain solute the concentration among the columnarThe ingot is in a rectangle shape of which the widthgrains is still higher due to the phase transition and theis1 mm,and the height is 2 mm. The ingot is meshedconcave part of columnar grain. Moreover ,the solute con-into 20000 cells and the size of each cell is 0.01 mm Xcentration in front of columnar tip is lower than those in0.01 mm. The bottom side of the ingot is heat-extractedother region around columnar zone ,so at the tip of colum-boundary where the heat-transfer cofficient is 500 W/nar the growth rate is much faster than that in the neigh-( m K ).The other side of the ingot is adiabatic boundary.borir中国煤化工- that the equiaxed grainsThe initial temperature of liquid is 1470 C and initial so-formrestrict a further growth oflute concentration is 0.6 % .The total calculating numberthe:TYHC N M H Gration rate is related withis 200000 steps.the local temperature based on Eq.( 24 ) ,the nucleation ismuch easier in the low temperature region , and in the di-4 Results and Discussionrection the lowest temperature region only lies ahead of thecolumnar tip .Fig. 3万施数揭e catculated microstructure morpholo-98Joumal of Wuhan University of Technology - Mater. Sci. Ed.Jun.2005.MIb)e)(fB)Fig.3 The microstructure evolution during solidifcation( a)(b)(c)(d)(e)( f){g){ h) are the 10000 ,50000 80000 100000 ,500000 ,180000 ,190000 200000 steps , respectively0.700.300.30. lmm0.1mm0. Imm0.lmm(d)a)(b)Fig.4 The solute concentration distribution during solidifcation process(a)(b){ c)( d ) are the 30000 , 10000 ,150000 and 190000 steps , respectivelySidebranching with Thermalnoise. Physical Rerier E ,1999 ,5 Conclusions60 3614-3625[3] Y T Kim ,N Provatas , N Goldenfeld ,J Dantzig. Computationof Dendritic Microstructure Using Level-set Method. Phys .a ) This model can describe the microstructure evolu-Rev. E 2000 62 2471-2474tion during the directional solidification process , as well[4] H S Udaykumar and L Mao. Shape Interface Simulation ofas columnar-to- equiaxed transition .Dendritie Sldifciton. Iu. J. Heat Mass Transfer ,2002 ,b) This model can reflect the solute redistribution45 :4793-4808and micro-segregation among the columnar grains and a-[5 ] Damir Juric and Gretar Tryggvason. A Front-tracking Methodlong the columnar from the stem center to the edge offor Dendrtie Soldifcaton. J. Compt. Plhys. ,1996 ,123 :columnar.127- 148c) The model has considered the effect of surface[6] M F Zhu J M Kim and C P Hong. Modeling of Globular andtension , which makes the tip of columnar become theDendritic Structure Evolution in Solidificaton of an Al-7mass% Si Alloy. ISI] Inter. 2001 A1 9929989thinnest part of the whole columnar .[7] L Nastac. Numerical Modeling of Solidificaiton Morphologiesand Segregation Pattems in Cast Dendrtic Alloy. ActaReferences[8]中国煤化工chel. Numerical Simulation of[ 1 ] Britta Nesler ,Adam A Wheeler and Harald Garcke . ModelingHCNMHGxdidied Cellular Automata. InMicrostructure Formation and Interface Dyamics. Computa-..... ... "Phenomena 3. ed. ,H ,Cerjaktional Materials Science 2003 26 111-119and H K D HBhadeshia sthe Institute of Materials ,the Univer-[2] Alsin Karmna and W outer-Jan Rappel. Phase- field of Dendriticsity of Cambridge ,JU K ,1997 85-105

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