numerical simulation of microstructure and solutal macrostructure

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Numerical simulation of microstructure andsolutal microsegregation formation of ternaryalloys during solidification process

Q. Li*1, Y. Wang1, H. W. Zhang, S. S. Xie2 and G. J. Huang2

A model of microstructure and microsegregation of ternary alloys is presented using a cellular

automaton (CA) method. In the model the solid fraction is deduced from the relationship between

the temperature, solute concentration and curvature of the solid/liquid interface, which can be

expressed as a cubic equation instead of assuming the solid/liquid interface position and shape,

so as to deduce the solid fraction according to the interface velocity. Then, using the model, a

dendrite of Fe–C–Si ternary alloy with 0 and 45u of preferential growth direction are simulated

respectively. Finally, a solidification microstructure and solute microsegregation are simulated,

and the simulated results can represent the microstructure and different solute segregation

during solidification process. Furthermore, the model can be extended to multicomponent alloys

solidification process.

Keywords: Simulation, Ternary alloys, Solute microsegregation

IntroductionA dendrite is a classic morphology of crystals duringsolidification, thus studying dendrite evolution is helpfulto understand the essence of solidification microstruc-ture and microsegregation. In the past decades manyresearchers used different methods to study dendriteevolution, such as the theoretical analysis method,1–4

Monte Carlo method,5 cellular automaton (CA)method,6 phase field,7 front tracking method8 and levelset method.9 Most of theoretical methods can only solvea one-dimensional problem, whereas the practicalproblems are always of two or three dimensions. TheMonte Carlo method can predict the trend of micro-structure evolution. Although the model can reflect thegrowth kinetic of dendrite tip, it cannot correspond toreal times. In the phase field method the meshed sizemust be smaller than the thinnest layer of interface,which limits the calculation efficiency. The fronttracking method is a sharp interface approach whichuses the discrete marked point to reflect the interfaceevolution. When the microstructure grows, more andmore marked points are adopted to track the interface,which increases its computational time. In addition, thefront track method is difficult to deal with mergingmicrostructures.

Compared with other methods, the CA method hasadvantages in that it has simple rules, without limitation

by mesh size, and need not track the topology of thesolid/liquid interface. Former researchers6,10,11 who usedthe CA method to simulate solidification microstructureonly took account of thermal diffusion without con-sidering solute diffusion effects. In 1997 Dilthy et al.12

presented a CA model for both thermal and solutaldendrites based on thermal and solutal diffusioncontrolled growth mechanism. In 1999 Nastac13 pre-sented a model which combined thermal diffusion withsolutal diffusion, and the model can simulate thecolumnar to equiaxed transition and show the interac-tion between the growing dendrites. In 2003, Beltran-Sanchez and Stefanescu14 modified the calculationmethod of the local solid/liquid interface curvature toreplace the method of counting the solid fraction of thenearest neighbour cells adopted by Dilthey et al. andNastac. 10–13 In 2004, Beltran-Sanchez and Stefanescu15

introduced the front track method into the CA modelthat can simulate different preferential growth dendrites.

All these variants of the CA method assumed thatthere was a sharp interface and the solid fraction in themushy zone was obtained according to the interfaceposition and interface velocity. However, the shape andposition of dendrite morphology is unknown, whichleads to the inaccurate solution of the solid fraction inthe solid/liquid interface unit. In order to overcome thisproblem, Jacot and Rappaz combined the Thermo-Calcsoftware with CA model to calculate the solid fraction inmulticomponent alloy solidification,16 which they calledthe pseudofront tracking method.16 This is similar to theCA method, in which the solid fraction of the solid/liquid interface unit is obtained by Thermo-Calc soft-ware according to the local solute concentration,temperature and curvature of the mushy zone. The

1School of Material Science and Engineering, Shenyang University ofTechnology, Shenyang, 110023, China2State Key laboratory for Fabrication and Processing of Non-ferrousMetals, General Research Institute for Non-ferrous Metals, Beijing,100088, China

*Corresponding author, email [email protected].

442

� 2009 Institute of Materials, Minerals and MiningPublished by Maney on behalf of the InstituteReceived 24 August 2008; accepted 10 November 2008DOI 10.1179/174328109X401604 Ironmaking and Steelmaking 2009 VOL 36 NO 6

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method can simulate not only binary alloy solidification,but also multicomponent alloy solidification.

The method used by Jacot presents an idea that thesolid fraction of the solid/liquid interface unit can beobtained by a calculated phase diagram method.Therefore, in this paper an improved CA model ispresented to calculate the solid fraction of the solid/liquid interface unit according to calculated phasediagram, in which the exact position of the interfaceposition is not required to be known, and the solidfraction of the interface unit can be obtained by solvinga quadric equation of the solid fraction in the solid/liquid interface unit instead of using Thermo-Calcsoftware. The essence of the model is that if the soluteconcentration, temperature and the curvature of arandom unit are known, then all these state parametersresponding to a point in the phase diagram can bededuced. Therefore, if the state parameters of any unitsare known, the solid fraction can be obtained accordingto the phase diagram. In order to illustrate the main ideaof the model, Fe–C–Si ternary alloy is adopted in thispaper.

Mathematical physical modelThe model is based on the following assumptions:

(i) the density of the ternary alloy is assumed as aconstant which is not changed during thesolidification process

(ii) there is a local phase equilibrium state in thesolid–liquid mushy zone

(iii) the influences of liquid convection are ignored

(iv) there is no chemical reaction between the twosolute components

(v) there are three kinds of phase states: solid,liquid and the solid/liquid interface in which thesolid fraction ranges from zero to one; inaddition, the solid/liquid interface unit must liebetween the solid and liquid units

(vi) the specific heat is a constant and the same forall three states.

Energy conservation equation

LT

Lt~+:(a+T)zQ (1)

where T is temperature, t is time, a is the thermaldiffusing coefficient, and a5k/(rCp), k is the conductiv-ity, r is the density, Cp is the specific heat and Q is thelatent heat released as the solid/liquid interface advancesinto the undercooled liquid, which can be given as

Q~Lfs

Lt

Lh

Cp

(2)

where fs is the solid fraction of the solid/liquid interfacezone, Lh is the latent heat released per volume.

Solute diffusion governing equation

LCi,I

Lt~+:(Di,Ii

+Ci,I) (3)

where D is the solute diffusing coefficient, the subscript Istands for the liquid or solid phase, the subscript i standsfor the different solute components.

When the solid/liquid interface advances, the solutewill be redistributed in the solid/liquid interface unit,which can be expressed as

C�1,s~k1,pC�1,l

C�2,s~k2,pC�2,l

((4)

where k1,p and k2,p are the solute partition coefficients ofthe different solute components, which are assumed as aconstant in this article, C�1,l and C�2,lare the liquid solute

concentration in the solid/liquid interface unit, C�1,s and

C�2,s are the solid solute concentration in the solid/liquid

interface unit, superscript 1 and 2 stand for the differentsolute components.

According to the phase diagram, if kinetic under-cooling is ignored, the relationship between the soluteconcentration, curvature and temperature in the solid/liquid interface unit is given by

T�~Teqzm1(C�1,l{C1,0)zm2(C�2,l{C2,0){

CK f(h,Q) (5)

where T* is the temperature of the solid/liquidinterface unit, Teq is the equilibrium temperature, m1

and m2 are the liquidus slope of components 1 and 2,which are assumed as a constant, C1,0 and C2,0 are theinitial solute concentration of components 1 and 2respectively, c is the Gibbs–Thomson coefficient, K is thecurvature of the solid/liquid interface, C�1,l and C�2,l are

the interface solute concentration of components 1and 2 respectively, f(h, Q) is the anisotropic functionof the solid/liquid interface, h is the normal directionof the solid/liquid interface and Q is the preferentialgrowth direction. f(h, Q) can be calculated according toRef. 14.

The curvature of the solid/liquid interface canbe deduced from the formula of curvature definition14,17

K~z’’

(1zz’2)3=2(6)

where z9 and z0 are the first and second derivativesof the solid fraction respectively, and which aregiven

z’~d y

d x~

d fs=d x

d fs=d y~

(fs)x

(fs)y

(7)

z’’~d y’d x

~d fs=d x

d fs=d y’

~(fs)

2x(fs)yyz(fs)

2y(fs)xx{2(fs)x(fs)y(fs)xy

(fs)3y

(8)

K~z’’

(1zz’2)3=2

~2(fs)x(fs)y(fs)xy{(fs)

2x(fs)yy{(fs)

2y(fs)xx

(fs)2xz(fs)

2y

h i3=2(9)

the subscript of x and y are the deviation of the solidfraction in x and y axis directions.

Equation (1) can be divided into two parts: in the firstpart, the latent heat is neglected and only the heatdiffusion is considered, which can be expressed asequation (10); in the second part the heat diffusion isneglected and the latent heat is considered only, whichcan be expressed as equation (11)

Li et al. Numerical simulation of microstructure and solutal microsegregation formation of ternary alloys

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LT

Lt~a+2T (10)

T ’~TzLh

Cp

dfs (11)

where T9 is the temperature in the solid/liquid interfaceunit that considered the latent heat released, T is thetemperature calculated by heat diffusing equation with-out considering the latent heat released, dfs is theincrement of the solid fraction in the solid/liquid

interface unit, dfs~f nz1s {f n

s . Therefore, equation (11)

can be rewritten as following

T ’~TzLh

Cp(f nz1

s {f ns ) (12)

where the superscript nz1 and n stand for the (nz1)thand the nth steps respectively.

For these interface units there are three kinds of soluteconcentration for each solute component – the liquidsolute concentration, the solid solute concentration andthe averaged solute concentration respectively. In thesolid phase, the average solute concentration is equal tothe solid solute concentration, and the liquid soluteconcentration is equal to zero; in the liquid phase theaverage solute concentration is equal to the liquid soluteconcentration, and the solid solute concentration isequal to zero; in the solid/liquid interface unit, the

average solute concentration is equal to the sum of thesolid fraction multiplying the solid solute concentration,and the liquid fraction multiplying the liquid soluteconcentration. Therefore the average solute concentra-tion is given as

Ci~

Ci,l fs~0

C�i,sfnz1

s zC�i,l(1{f nz1s ) 0vfsv1

Ci,s fs~1

8><>: (13)

where C is the averaging solute concentration at the(nz1)th step, the subscript l and s are corresponded tothe liquid and solid phases respectively.

As the solid and the liquid within these units areapproximated to be of uniform composition, the liquidsolute concentration in the solid/liquid interface unit canbe deduced as follows

Ci�l ~

Ci

1{f nz1s zki,p

:f nz1s

~Ci

1{(1{ki,p):f nz1s

(14)

Combined the Eq. (5) and Eq. (11), the followingrelationship can be got:

T ’~T�~TzL

Cp(f nz1

s {f ns )

~Teqzm1(C1�1,l{C1,0)zm2(C�2,l{C2,0){CK f(h,Q)

(15)

Brought equation (14) into (15), it can be drawn

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444 Ironmaking and Steelmaking 2009 VOL 36 NO 6

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TzLh

Cp

(f nz1s {f n

s )~

Teqzm1C1

1{(1{k1,p):f nz1s

{C1,0

� �z

m2C2

1{(1{k2,p):f nz1s

{C2,0

� �{CKf(h,Q) (16)

1{f nz1s (1{k1,p)

� �1{f nz1

s (1{k2,p)� �

is multiplied in

both sides of equation (16), and then the followingequation can be obtained

T{Teq{Lh

Cpf ns zm1C1,0zm2C2,0zCK f(h,Q){

m1C1{m2C2

z

Teq{Tz Lh

Cpf ns {m1C1,0{m2C2,0{

CK f(h,Q)zm1C1(1{k2,p)z

m2C2(1{k1,p)

2664

3775:f nz1

s

z

T{Teq{Lh

Cpf ns zm1C1,0z

m2C2,0zCK f(h,Q):(1{k1,p):(1{k2,p)z

(2{k1,p{k2,p): Lh

Cp

2664

3775:(f nz1

s )2

z LCp

:(1{k1,p):(1{k2,p):(f nz1s )3~0

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

(17)

Equation (17) can be simplified as follows

A:(f nz1s )3zB:(f nz1

s )2zC:(f nz1s )zD~0 (18)

where A, B, C and D are coefficients and constant, whichcan be expressed as

A~ Lh

Cp

:(1{k1,p):(1{k1,p)

B~T{Teq{Lh

Cpf ns zm1C1,0zm2C2,0z

CK f(h,Q):(1{k1,p):(1{k2,p)z

(2{k1,p{k2,p): Lh

Cp

C~Teq{Tz Lh

Cpf ns {m1C1,0{m2C2,0{

CK f(h,Q)zm1C1(1{k2,p)zm2C2(1{k1,p)

D~T{Teq{Lh

Cpf ns zm1C1,0zm2C2,0z

CK f(h,Q){m1C1{m2C2

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

(19)

where C1,0 is the initial liquid carbon concentration, andC1,050?6%; C2,0 is the initial liquid silicon concentra-tion, and C2,050?15%.

Equation (18) is a classic cubic equation, which hasthree roots at most. In this paper, the Newton iterationmethod is adopted to solve equation (18), which can beexpressed as

fs~fs{y=y’ (20)

where y~A:(f nz1s )3zB:(f nz1

s )2zC:(f nz1s )zD and y9 is

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the first order difference for f nz1s , and y’~

3:A:(f nz1s )2z2:B:(f nz1

s )zC. Therefore, equation (18)

has three roots at most, and the reasonable root should

be satisfied with the relationship f nz1s {f n

s

�� ¡0:2 in each

time step, that is to say, in each time step, the solidfraction increment is not more than 0?2.

The boundary condition and the initial conditions aregiven as following

LT=Lt~h(T{Tsur)

T(x,y,t~0)~T0

C(x,y,t~0)~C0

8><>: (21)

where h is the interface heat transferring coefficient on

the boundary, Tsur is the surrounding temperature,

which is assumed as a constant and Tsur525uC and T0 is

initial liquid temperature.

In this model, the interface velocity cannot be deducedfrom the solid fraction because the interface shape andposition is unknown. Therefore, the advancing velocityof the solid/liquid interface can be given as

V~ Di,s

LCi,s

Ln

� �s

{Di,l

LCi,l

Ln

� �l

� �= C�i (1{ki,p)� �

(22)

The time step is an important factor affecting thecalculating efficiency, which can be calculated according

to following equation

dt¡1

5Min

a2

D1,l

,a2

D1,s

,a2

D2,l

,a2

D2,s

,a2

a,

a

Vmax

� �(23)

where Min is a function of selecting the minimum value

among a2

D1,l, a2

D1,s, a2

D2,l, a2

D2,s, a2

a and aVmax

, dt is time step, Vmax

is the maximum velocity in the all solid/liquid interfaceunit, which can be obtained from equation (22), a is thesize of the meshed grid, 1/5 means that the advancinginterface cannot move over 1/5 of the meshed grid ineach time step.

The CA rules of nucleation and capturing are adoptedas Refs. 13 and 14, and the solute diffusing equation canbe solved according to Ref. 14.

The calculation steps are given as follows:(i) setting the initial and boundary conditions

according to equation (21)(ii) calculating the time step according to equa-

tion (23)(iii) calculating the temperature field without con-

sidering the latent heat releasing according toequation (10)

(iv) calculating the solid fraction according toequation (18)

(v) Bringing the new solid fraction of the solid/liquid interface unit into equations (9), (11),

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(14) and (22) to get the curvature, temperaturesolute concentration and velocity of the solid/liquid interface unit

(vi) calculating the solute concentration accordingto equation (3) that takes the change of thesolute concentration in the solid/liquid interfaceunit into account

(vii) restoring the variable data for each unit, such astemperature, solute concentration, curvatureand solid fraction

(viii) repeating the above steps until reaching theappointed calculating number.

In this paper an Fe–6C–0?15Si ternary alloy was

adopted, and the parameters used are listed in Table 1.The size of meshed grid is 161 mm, and the whole

simulated zone is 0?2560?25 mm.

The silicon solute diffusion coefficient in the solid and

liquid phase is assumed to be the same as carbon, andthe liquid slope and solute partition coefficient are

selected by calculating phase software.

In order to simplify the model, first a solid seed is put

into the centre of the simulated zone with a uniformtemperature, whose preferred growth direction adopts 0

and 45u, respectively. Then, a stochastic nucleation

model is adopted to simulate the solidification micro-

structure and solute microsegregation in the same

solidification conditions.

Results and analysisFigure 1 indicates that a solid seed grows as thesolidification time increases. The colour bar in the rightside of Fig. 1 corresponds to the different carbon soluteconcentration. When the solidification time is 0?002 s,the solid seed prefers to grow along the vertical andhorizontal directions, which can be seen in Fig. 1a, andthe carbon concentration of the nucleus is about 0?4%.When the solidification time is 0?08 s, the solid seedbecomes a dendrite, as shown in Fig. 1b, and the carbonconcentration in the nucleus increases to 0?5% becauseof the back diffusion in the solid phase. In Fig. 1c and d,the dendrite grows, and the carbon concentration in theprimary dendritic arm also rises gradually due to soluteback diffusion in the solid phase. When the solidificationtime is 0?016 s, the secondary dendrite arms appearalong the primary dendritic arms because of under-cooling caused by the carbon concentration. In addition,Fig. 1 shows that the carbon concentration in theconcave part of the dendrite contour is higher than thatin the protruding part of the dendrite contour.

Figure 2 indicates the silicon concentration distribu-tion during solidification process. At the solidificationtime of 0?002 s (Fig. 2a), the silicon concentration in thesolid seed is about 0?095%. When the solidification timeis 0?008 s, the silicon solute concentration is about0?125% in the primary dendrite arms except for the

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centre of the dendrite. This is caused by the low siliconsolute diffusion coefficient. When the solidification timeis 0?012 s, the dendrite grows continuously, and thesilicon solute concentration at the tip of primarydendrite arm approaches 0?14%. When the solidificationtime is 0?016 s, the second dendrite arms grow and thesilicon solute concentration at the edge of the primarydendrite arms is nearly 0?14%. Comparing Fig. 2 withFig. 1, it can be seen that at the same solidification timethe dendrite morphology is also same, but the silicon

solute concentration distribution along the primarydendrite arms is much more uneven in Fig. 2 thancarbon solute concentration distribution along primarydendrite arms in Fig. 1.

Figure 3 indicates the carbon concentration distribu-tion. When the solidification time is 0?002 s (Fig. 3a),carbon concentration in the solid seed is about 0?4%.When the solidification time is 0?008 s (Fig. 3b), thesolid seed begins to grow and its preferential growthdirection is 45u from the horizontal direction.

Table 1 Parameters used in model

Density, kg m23 7300*Heat diffusion coefficient, m2 s21 6.1E-6Carbon solute diffusion coefficient in liquid phase, m2 s21 2E-9*Carbon solute diffusion coefficient in solid phase, m2 s21 5E-10*Carbon solute partition coefficient at equilibrium condition 0.34*Silicon solute diffusion coefficient in liquid phase, m2 s21 2E-9Silicon solute diffusion coefficient in solid phase, m2 s21 5E-10Silicon solute partition coefficient at equilibrium condition 0.59Latent heat, kJ kg21 270*Specific heat, J kg21 K21 800*Liquidus slope in equilibrium condition of Fe–C phase diagram, K/% 280*Liquidus slope in equilibrium condition of Fe–Si phase diagram, K/% 217.1Gibbs–Thomson coefficient, m K21 1.9E-7*Melting temperature, uC 1490*Initial temperature, uC 1480*Boundary heat transfer heat coefficient, W m22 K21 1000

*Parameters adopted from Ref. 13.

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Furthermore, the carbon concentration in the dendritearms also increases to about 0?5%, which can be seen inFig. 3b. When the solidification time is 0?012 s (Fig. 3c),the dendrite grows continuously and the second dendritearms appear along the primary dendrite arms. When thesolidification time is 0?019 s (Fig. 3d), the dendritegrows continuously, and the carbon concentration inthe local zone approaches 0?55%. In addition, Fig. 3shows clearly that there are more secondary dendritearms forming along the primary dendrite arms than inFig. 1, which leads to the solute accumulated zoneforming among the dendrite arms.

Figure 4 indicates the silicon concentration distribu-tion at different solidification times. When solidificationtime is 0?002 s, as shown in Fig. 4a, the solid seed grows,and the silicon concentration in the solid seed is about0?11%. When solidification time is 0?08 s (Fig. 4b), thesolid seed turns into a dendrite, in which the preferentialgrowth direction is 45u, and the silicon concentration ofthe dendrite increases to 0?125%. As the solidificationtime reaches 0?012 s (Fig. 4c), the second dendrite armsappear along the primary dendrite arms, and the siliconconcentration in the primary dendrite arm is near to0?14% and the silicon concentration among the second-ary dendrite arms is more than 0?185%. At thesolidification time of 0?019 s (Fig. 4d), the siliconconcentration in the primary dendrite arms and the

second dendrite arms are about 0?14 and 0?125%,respectively. In addition, comparing Fig. 4 with Fig. 3,it can be seen that the morphology of dendrites is thesame at the same solidification time.

Figure 5 is the carbon concentration distribution atdifferent solidification times. When the solidificationtime is 0?002 s, some nuclei form randomly in the undercooling liquid, as shown in Fig. 5a, and the carbonconcentration in these nuclei is less than 0?45%. At asolidification time of 0?006 s (Fig. 6b), some nucleiprefer to grow parallel to the horizontal direction andothers prefer to grow at 45u. When the solidificationtime is 0?012 s (Fig. 6d), some dendrites grow continu-ously until they meet one each other or reach theboundary. When the two dendrites meet, there will be acarbon enriched zone between the two primary dendritearms, which leads to the carbon microsegregation on thegrain boundary after the liquid solidifies completely.

Figure 6 shows silicon distribution at different solidi-fication times, which reflects the morphology evolutionof solidification microstructure during solidification. Ata solidification time of 0?002 s (Fig. 6a), some nucleiform randomly in the undercooled liquid, and the siliconconcentration in the nuclei is about 0?11%. When thesolidification time is 0?006 s (Fig. 6b), the nuclei grow,the dendritic morphology appears, and the siliconconcentration in the dendrites is about 0?11%. As the

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solidification time increases, as can be seen in Fig. 6c,these dendrites grow continuously and the siliconconcentration in these dendrites rises to about 0?125%.When the solidification time is 0?012 s (Fig. 6d), thedendrites grow continuously until they meet each otheror they meet the boundary. From Fig. 6, it can be seenthat the secondary dendrite arms are easier to growexcept when the preferential growth direction is 0u.

ConclusionsA new model is presented to simulate dendrite evolution,solute microsegregation and solidification microstruc-ture. The following conclusions can be drawn from thesimulation results.

1. Owing to the back diffusion effect, solute concen-tration in the solid phase gradually increases as solidifi-cation proceeds and solute concentration at the dendriteedge is higher than at the dendrite centre. The last liquidto solidify contains the highest solute concentration.

2. Owing to the different solute partition coefficientsof C and Si, as solidification proceeds, Si concentrationdistribution along the primary dendrite arm is moreuneven than C.

3. When two dendrites meet, there is a solute enrichedzone between them, which leads to the solute concentra-tion and solidification rate decrease.

4. Side branch dendrite growth is more prevalentwhen the primary dendrite growth direction is notparallel to the heat diffusion direction.

5. The model can be extended for use in multi-component alloy solidification as well as ternary alloy.This work is currently underway.

Acknowledgements

The work was supported by Liaoning EducationalDepartment (grant no. 05L-304), and supported by theChinese Post Doctoral Foundation (grantno. 2008043340).

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numerical simulation of microstructure and solutal macrostructure

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