Computer Modeling of the Evolution of Dendrite Microstructure in …rm/PDF/mica2.pdf ·...

Computer Modeling of the Evolution of DendriteMicrostructure in Binary Alloys During Non-isothermal Solidification

M. Grujicic, 1,2 G. Cao,1 and R. S. Miller1

Two-dimensional simulations of the evolution of dendrite microstructure during isothermal andnon-isothermal solidifications of a Ni-0.41Cu binary alloy are carried out using the phase-fieldmethod. The governing evolution equation for the phase field variable, the solute mole fraction andthe temperature are formulated and numerically solved using an explicit finite difference scheme.To make the computations tractable, parallel computing is employed. The results obtained showthat under lower cooling rates, the solidification process is controlled by partitioning of the solutebetween the solid and the liquid at the solid/liquid interface. At high cooling rates, on the otherhand, solute trapping takes place and solidification is controlled by the heat extraction rate. An in-crease in the cooling rate is also found to have a pronounced effect on the dendrite microstructurecausing it to change from poorly developed dendrites consisting of only primary stalks, via fullydeveloped dendrites containing secondary and tertiary arms to the diamond-shaped grains with cel-lular surfaces. These findings are in excellent agreement with experimental observations.

KEY WORDS: Phase-field method; solidification modeling; parallel computing.

computational space without making distinction be-tween different phases. This approach enables simula-tions of the microstructure evolution without explicittracking of the phase interfaces and does not entail theuse of a priori assumptions about the evolution paths.This, in turn, makes microstructure-evolution simula-tions computationally very efficient in comparison totheir traditional sharp phase-interface counterparts. Asfar as the relations between the phase-field and sharp-in-terface approaches are concerned, Caginalp et al. [3, 4]demonstrated that the governing phase-field equationsreduce to the corresponding sharp-interface equations inthe limit of a zero interface thickness and that the resultspredicted by the phase-field model are quite close to thecorresponding sharp-interface results, even for relativelylarge values (,50 nm) of the interface thickness.

Despite the aforementioned advantages offered bythe phase-field method, its application to modeling micro-structure evolution during the solidification of metallicmaterials has been so far limited to relatively simple sys-tems (e.g., to non-isothermal solidification of pure metals[1] or isothermal solidification of binary alloy systems

1911064-7562/02/0700-0191/0 © 2003 Plenum Publishing Corporation

Journal of Materials Synthesis and Processing, Vol. 10, No. 4, July 2002 (© 2003)

1 Department of Mechanical Engineering, Clemson University, Clem-son, SC 29634-0921.

2 To whom correspondence should be addressed. Tel: 864-656-5639;Fax: 864-656-4435; e-mail: [email protected]

1. INTRODUCTION

Over the last decade, the phase-field method hasbeen increasingly used to model liquid→solid [1] andsolid→solid [2] phase transformations and the accompa-nying microstructure evolution. Within this method, anew variable denoted as the phase-field variable (f) is in-troduced to represent the physical (structural) state of thematerial at each point. The phase-field variable takes ondifferent fixed values in different crystal phases (e.g., foran alloy solidification problem, f 5 0 in the solid andf 5 1 in the liquid are generally used) and changessharply but smoothly over a thin layer which separatesthe phases. This layer plays the role of classical phase in-terfaces. An evolution equation for the phase-field vari-able is next developed, coupled with (phase-field modi-fied) transport equations and applied to the entire

obeying the ideal solution thermodynamics [1]). Alloys ofthe technological importance, however, are typically multi-component with complex thermodynamics interactions be-tween the species and solidify over a temperature rangeduring casting. Thus, for the phase-field method to achieveits full potential for predicting microstructure evolution, itsapplication must be extended to the technologically impor-tant materials and to more realistic solidification process-ing conditions. The present paper represents an initially at-tempt to extend the phase-field method in these directions.

The starting point for the present work is the two-dimensional phase-field model of Warren and Boettinger[1] for isothermal solidification involving dendritegrowth of highly supersaturated binary melts. The modelwas subsequently extended to include the effect of heatflow resulting from release of the latent heat [6, 7]. Thenumerical implementation of this method is found to bequite challenging primarily due to the fact that the tem-perature and the solute concentration evolve at very dif-ferent time scales. To overcome this problem, Boettingerand Warren [8] proposed a simplified model withinwhich the spatial variation of the temperature is neg-lected and the temperature evolution equation replacedby a heat balance equation involving the imposed heatextraction rate and the latent-heat release rate.

In the present work, parallel computing is used tohandle the computational demands associated with mod-eling the non-isothermal solidification of binary alloy sys-tems brought about by time-scale differences in the tem-perature and the solute concentration evolutions. Inaddition, the effect of solvent-solute atomic interactions istaken into account in the thermodynamic models for boththe solid and the liquid. The organization of the paper isas follows: In Section 2.1, a brief overview is given of thephase-field model used in the present work. Details asso-ciated with numerical solutions of the governing differen-tial equations using parallel computing are given in Sec-tions 2.2 and 2.3. The main results obtained in the presentwork are presented and discussed in Section 3. The keyconclusions resulted from the present work are listed inSection 4.

2. COMPUTATIONAL PROCEDURE

2.1. The Phase-Field Model

The phase-field model used in present work is basedon the following entropy functional [9]:

(1)S 5 #V

as(f, xB, u) 2«2

2ƒ §f ƒ 2bdV

where s is the thermodynamic entropy density, f thephase-field variable which varies smoothly between 0 inthe solid and 1 in the liquid, xB the mole fraction of soluteB (Cu in the present work) in solvent A (Ni in the presentwork), u the internal energy density and V the materialvolume. The interfacial energy parameter « is assumed tobe anisotropic (i.e., dependent on the crystallographic ori-entation of the solid/liquid interface) as:

(2)

where g is the anisotropy amplitude, k the mode numberand v 5 arctan (fy/fx) is the angle between the local in-terface normal and a designated base vector of the crys-tal lattice. Subscripts x and y are used to denote partialderivatives with respect to these two spatial coordinates.The designated base vector of the two-dimensionalsquare crystal lattice is typically aligned with the x-axis.For each of the pure species, the parameter ¯« in Eq. (2)is defined as [10]: where is j the solid/liquid interface energy, d the interface thickness and Tm

the melting point.Since the phase-field variable f is not a conserved

quantity, the simplest form of its evolution equation, whichguarantees that the total entropy of the system increasesmonotonically during solidification, can be defined as [10]:

(3)

where the raised dot is used to denote a time derivative,Mf is an interfacial mobility parameter and the operatord is used to denote a variational derivative.

The evolution of the other two dependent variables,xB and u, is given by the normal conservation laws as:

(4)

and

(5)

where ¹ ? is the divergence operator, Vm the molar vol-ume and JB and Ju are the diffusional flux and the heatflux, respectively.

Application of the df variational derivation to Eq. (1)yields:

(6)

where and ¹ is the gradient operator.h¿ 5 dh/dv

1 « 2

yahh¿

f

yb

dS

df5

s

f1 « 2§(h2§f) 2 « 2

xahh¿

f

xb

?u 5 2§ ? Ju

?xB

Vm5 2§ ? JB

?f 5 Mf

dS

df

«2 5 622jd/Tm

« 5 «h 5 «(1 1 g cos kv)

192 Grujicic, Cao, and Miller

As shown [1], the first term on the right hand sideof Eq. (6) can be expressed as:

(7)

where f (f, xB, T) 5 [(1 – xB)mA(f, xB, T) 1 xBmB(f, xB,T)]/Vm is the Helmholtz free energy density of thealloy while mA(f,xB,T) and mB(f,xB,T) are the corre-sponding chemical potentials of the two constituentspecies.

Within the regular solution framework, the twochemical potentials are defined as:

(8a)

and

(8b)

where fA(f, T) andfB(f, T) are phase-field variable mod-ified Helmholtz molar free energy densities for the twoconstituent species, VAB(f) the regular solution interac-tion parameter and R the universal gas constant.

Using the standard thermodynamic relationship:fI(f, T) 5 uI((f, T 2 TsI(f, T)) and defining the internalenergy density for each constituent species as:

(9)

Warren and Boettinger [1] showed that:

(10)

where is the melting point of the species I, cI is the spe-cific heat (assumed to be the same in the solid and inthe liquid), p 5 f3(10 215f 1 6f

2) a function whichsmoothly connects the solid (f 5 0) state with the liquid(f 5 1) state, (subscripts L and Sused to denote the liquid and the solid states) is the latentheat of fusion, GI(f) 5 WIg(f) 5 WIf

2(1 2 f)2 is adouble-well function (with a WI barrier height) which en-sures that the Helmholtz free energy density minima arelocated in the solid (f 5 0) state and in the liquid (f 5 1)state at each temperature.

Through the use of Eq. (10), Eq. (7) can be ex-pressed as:

(11)2 xB(1 2 xB)V¿AB(f)/(TVm)

s

f5 2(1 2 xB)HA(f, T ) 2 xBHB(f, T )

LI 5 uI,L(T Im)2uI,S(T

Im)

TIm

a1 2T

T Im

b 2 cIT lnT

TIm

(I 5 A, B)

fI 5 GI(f)T 1 CuI,S1TIm 2 2 cIT

Im 1 p(f)LI D

uI 5 uI,S(TIm) 1 cI(T 2 TI

m) 1 p(f)LI (I 5 A, B)

1 RT lnxB

mB(f, xB, T ) 5 fB(f, T )Vm 1 VAB(f)(1 2 xB)2

1 RT ln(1 2 xB)mA(f, xB,T ) 5 fA(f,T )Vm 1 VAB(f)x2

B

s

f5 2

1

T

f

f5 2

1

TVm

f C 11 2 xB 2mA 1 xBmB D

where

(12)

while V9AB(f) 5 p9(f)(VAB,L 2 VAB,S) and g9(f) 5dg(f)/df is used to denote the first derivative of g(f).

Upon substitution of Eqs. (2) and (11) into Eq. (6),carrying out the partial differentiations in the last twoterms in Eq. (6) and after substitution of Eq. (6) intoEq. (3) one obtains:

(13)

where ¹2 denotes the Laplacian operator, the interfacialmobility parameter for the alloys is defined as: Mf 5(1 2 xB)MA,f 1 xBMB,f and:

(14)

where b is the linear kinetic coefficient which relates in-terface velocity vi to the interface undercooling as:

, (I 5 A,B. Eq. (13) represents the finalform of the evolution equation for the phase-field variable.

The diffusional flux JB and the heat flux Ju, appear-ing in Eqs. (4) and (5), are given by the irreversible ther-modynamics linear laws as:

(15)

and

(16)

where the parameters LBB and Luu are related to the A-Binter-diffusion coefficient and the heat conductivity, re-spectively. It should be noted that the contributions frominteractions between the species and the internal energytransports have been neglected in Eqs. (15) and (16).

Using the previously given relation between theHelmholtz free energy density and the chemical poten-tials and assuming that the internal energy density u andthe entropy density sare weak functions of temperatures,

Ju 5 Luu§dS

du

JB 5 LBB§dS

dxB

vIi 5 bI(T

Im 2 T)

MI,f 5TI 2

m bI

622LIdI

(I 5 A, B)

2 ¹2f 2 cos(2v)(fxx 2 fyy)]62

1

2 «[h¿2 1 hh9] 3 [2 sin(2v)fxy

1 «2hh¿[sin(2u)(fyy 2 fxx) 1 2cos(2u)fxy]

xB(1 2 xB)V¿AB(f)/(TVm)

?f 5 Mf5[ «2h2§2f 2 (1 2 xB)HA 2 xBHB 2

(I 5 A, B)

HI(f, T ) 5 WIg¿(f) 1 30g(f)LIa1

T2

1

T Im

b

Computer Modeling of the Evolution of Dendrite Microstructure 193

the two variational derivatives appearing in Eqs. (14) and(15) can be expressed as:

(17)

and

(18)

Substitution of Eqs. (8a), (8b), (10), (12) (17) intoEq. (15) yields:

(19)

A comparison of Eq. (19) with the Fick’s first lawin a single-phase system (¹f 5 0) establishes:

(20)

where D is the A-B inter diffusion coefficient. In addition,substitution of Eqs. (19) and (20) into Eq. (4) yields:

(21)

where

(22)

Equation (21) represents the final form of the evo-lution equation for the mole fraction of the solute.

To derive the evolution equation for temperature,the internal energy density of the alloy is first expressedusing a rule of mixture as:

(23)

Then, substitution of Eqs. (9), (16), (18) and (23) intoEq. (5) and setting Luu 5 KT2 where K is thermal con-ductivity yields:

(24)cT? 1 30g(f)Lf?

5 § ? K§T

u 5 (12xB)uA1xBuB

2 DS(xB,T)D(f, xB, T) 5 DS(xB, T) 1 p(f)(DL(xB, T)

(HB(f, T) 2 HA(f, T))§f d

?xB 5 § ? D(f, xB, T ) c §xB 1Vm

RxB(1 2 xB)

LBB 5 D xB(1 2 xB)

RVm

1 (HA(f, T ) 2 HB(f, T))§f dJB 5 LBB c2 R

xB(1 2 xB)§xB

dS

du5

s

u5

1

T

dS

dxB5

s

xB5 2

1

T

f

xB5 2

mB 2 mA

TVm

where c 5 (12xB)cA1xBcB, L 5 (12xB)LA1xBLB, and K 5(12xB)KA1xBKB. Eq. (24) represents the final form of theevolution equation for the temperature.

Numerical solution of the three governing evolutionequations (Eqs. (13), (21) and (24)) is discussed in nextsection.

2.2. Numerical Procedures

Before carrying out numerical solution of the threegoverning differential equations, the independent and de-pendent variables, as well as the model parameters,are non-dimensionalized by introducing the followingdimensionless quantities (denoted by a tilde):

(25)

where In order to enable an analysis of the dendrite mi-

crostructure at a characteristic length scale typically en-countered in conventional casting, a 100 mm by 100 mmtwo-dimensional square-shaped computational domain isselected. To reduce the computational time, two-foldsymmetry is assumed about the x and y axes, so that onlyone (the upper right) 50 mm by 50 mm quadrant had to beanalyzed. It should be noted that in order to make simu-lations of the solidification process physically realistic,the interface thickness d should be set to a value compa-rable with the capillary length (d0 5 j/L < 2 3 10210 m).If the mesh resolution is then made comparable with thecapillary length, the 50 mm by 50 mm computational do-main would contain between 1010 and 1011 points makingthe computations intractable. To overcome this problem,all the calculations presented here are carried out using alarger fixed value for the phase interface thickness, d 5 53 1028 m. The results obtained in the present work butnot shown here for brevity proved that the solid/liquid in-terface velocity (perhaps the most critical quantity pre-dicted by the model) is a weak function of the interfacethickness (e.g., a change in the interface thickness by afactor of 20 caused only a 25% change in the interface ve-locity). Thus the use of an interface thickness substan-tially larger than the capillary length appears justified.

The three governing differential equations, Eqs. (13),(21), and (24), are solved using a second-order accurate

T* 5 (TAM 1 TB

M)/2.

M~

I 5MId

2RT *

DLVM, «

~5

«

dB Vm

RT *

W~

I 5WIVm

RT *

x 5 dx~, y 5 dy~, t 5d2c

Kt~, T 5 T * T~, f 5

RT *

Vmf~,


explicit central-difference computational scheme. To en-sure stability of the explicit scheme, a dimensionless timestep Dt~ of 1.5.

Obtaining the solution for the present phase-fieldmodel in order to find time advancement of the three two-dimensional arrays, f(x,y,) xB(x,y), and T(x,y) would becomputationally quite expensive using single-processorcomputers of the type commonly available. For example,for a moderate discretization of a square domain using thenumbers of grid points in the x and y direction Nx 5 Ny 51500 (the total number of grid points 2,250,000), a typicalexecution time for one-time step of ,0.5 sec on a SUNULTRA 5 machine, and a typical number of simulationsteps of 10,000,000,it takes around 58 days to completeone run. Clearly, such simulation times are not acceptable.To overcome this problem, parallel computing with 25processors is utilized in the present work. Parallelizationof the code is done using a typical two-dimensional de-composition of the 50 mm by 50 mm square computationaldomain into 25 square subdomains of equal size. Eachprocessor is next assigned one such subdomain. Partition-ing of a portion of the computational domain into subdo-mains is shown schematically in Figure 1(a). In this fig-ure, each processor (denoted by a number) is assigned asubdomain (delineated using solid lines) and, for simplic-ity, each domain is assumed to contain 16 grid points. Theonly required interaction between the processors involvesdata exchange at the boundary between neighboring sub-domains needed to calculate local spatial derivativesacross the boundary. To enable such data exchange, one

row/column of additional grid points (referred to as the“ghost” points and denoted using open circles in Figure1(a)) is added to each edge of all subdomains. The ghostpoints in one subdomain are, in fact, the boundary pointsof the neighboring subdomains. Thus, values of the de-pendent variables at the ghost points of one subdomain areupdated by processors other than the one handling thenon-ghost points of the same subdomain and to ensure acomplete consistency between one-processor and multi-processor simulations, values of the dependent variables atthe boundary points of one subdomain are passed to theghost points of the neighboring subdomains once duringeach simulation step. The code for the phase-field modelis written in Fortran 77 and the communication along theboundaries of adjoining subdomains performed using theMessage Passing Interface (MPI) parallel communicationroutines. Parallelization of the phase-field model proved tobe very efficient resulting in near linear decrease of thecomputation time with an increase in the number ofprocessors for the first 16 processors, Figure 1(b). How-ever, since the 32-proceeor “Beowulf” cluster used in thepresent work consists of 16 dual-processor nodes, the lin-ear scaling could not be maintained above 16 processors.Typically, the simulation time using 25 processors was re-duced by a factor of ,22.

2.3. Problem Statement

The phase-field model developed in Section 2.1 isused to study the evolution of dendrite microstructure


Fig. 1. (a) Partitioning of the computational domain using solid lines into subdomains containing mesh points(solid circles) at which dependent variables are updated by the same processor (denoted by a number).Dependent variables at the ghost points (open circles) are assigned to each extended subdomain (the dashedsquare) and used in the computation of spatial derivatives. (b) A typical parallel-code speed-up as a functionof the number of processors. The speed-up is defined as the ratio of simulation times on one processor to thaton N processors. The solid line represents a perfect linear scaling.

during solidification of a Ni-Cu alloy containingmole fraction of copper from a melt

undercooled by dT 5 20.5 Kfrom the liquidus tempera-ture to a temperature of Tinit 5 1574 K.The correspon-ding liquid supersaturation defined as a ratio of the alloycomposition and the equilibrium liquid com-position is estimated as ,0.85. The equi-librium Ni-Cu phase diagram is shown in Figure 2.

The computational domain is assumed to be initiallyuniform relative to the mole fraction of copper( ) and with respect to the temperature(Tinit 5 1574 K). In addition, the material is assumed tobe in the liquid state except for a very small quarter cir-cular region of radius r0 (50.0667mm) in the bottom leftcorner of the 50 mm by 50 mm computational domain.This region is assumed to contain the solid with the molefraction of copper equal to . The initial condition forthe phase-field variable is defined by extending the solu-tion of a one-dimensional phase-field evolution equationfor isothermal coexistence of the solid and the liquid ina pure metal [10] to a two-dimensional case as:

(26)

The Neumann-type (gradient) boundary conditionsare used for all three dependent variables, f, xB, and T,over all four edges of the 50 mm by 50 mm computational

finit 5 (x, y, t 5 0) 51

2c1 1 tanh

2x2 1 y2 2 r20

d222d

x0Cu

xinitCu 5 x0

Cu 5 0.41

xeqCu,L 5 0.48

x0Cu 5 0.41

x0B 5 x0

Cu 5 0.41domain. The gradients are set to zero for f and xB overall four edges and for T only over the symmetry (bottomand left) edges. The values for the temperature gradientalong the outer (top and right) edges of the computa-tional domain are computed using the values for thecooling rate T? externally imposed along these edges anda simple heat-flux balance relation as:

(27)

where Lx is the edge length of the computational domain(550 mm).

To promote the development of secondary and terti-ary dendrite arms during solidification, a stochastic noiseterm (2Mfa?r ?16g(f)[(12xB)HA 1 xBHB]) is added tothe right hand side of Eq. (13), where r is a random num-ber distributed uniformly in the [21, 1] range, and it isgenerated for every grid point at each time step anda(50.4) is the fluctuation amplitude. The stochastic noiseterm is used to mimic temperature fluctuations which areparticularly pronounced at the solid/liquid interfaces. The16g(f) part of the noise term defines a double-well func-tion whose magnitude is zero at f 5 0.0and f 5 1.0andone at f 5 0.5 and hence ensures that fluctuations areconcentrated at the solid/liquid interface.

3. RESULTS AND DISCUSSION

3.1. Case A: Solidification under IsothermalConditions

In this section, the phase field model developed inSection 2.1 is used to study the isothermal solidificationof the Ni-0.41Cu alloy from a supersaturated liquid at1574 K.Toward that end, the temperature evolution equa-tion, Eq. (24), is not considered and, consequently, a sig-nificantly larger time increment (based on the solute-dif-fusion Courant number defined as d2/DL 5 2.5 3 1026 s)is used. Simulations of the isothermal solidification arecarried out in order to obtain a reference dendrite mi-crostructure with which dendrite microstructures ob-tained under different non-isothermal solidification con-ditions can be compared.

The copper mole fraction fields in the Ni-0.41Cualloy at four different simulation times (0.75 ms, 1.5 ms,2.25 ms, and 3.0 ms) during isothermal solidification areshown in Figures 3(a)-(d), respectively. At the shortestsimulation time (0.75 ms), Figure 3(a), the initially cir-cular solid particle is seen to have grown in size and tohave acquired a four-pointed star-like shape. It should be

§T 5 2cT?Lx

2K


Fig. 2. The Ni-Cu alloy phase diagram. The initial condition of thealloy is represented by point A while the corresponding equilibriumchemical compositions of the liquid and the solid are represented bypoints B and C respectively.

noted, as stated earlier, that only one (the upper-right)quarter of the solid particle is displayed in Figure 3(a),as well as in the subsequent figures. The results dis-played in Figure 3(a) show that partitioning of the cop-per between the solid and the liquid takes place at thesolid/liquid interface and that, as expected, the diffusionlength is significantly larger in the liquid.

Figure 3(b) shows that at a longer solidification time(1.5 ms), the solid/liquid interface becomes unstable withrespect to small perturbation caused by the introductionof the noise term in the differential equation for the phasefield variable, Eq. (13). Consequently, secondary den-drite arms begin to form and grow into the undercooledliquid. Copper partitioning between the solid and the liq-uid continues and the diffusion distance in the solid hasincreased.

Figure 3(c) shows that as the solid particle contin-ues to grow between 1.5 ms and 2.25 ms, additional sec-

ondary dendrite arms form and grow into the under-cooled liquid. Diffusion of the copper into the solid con-tinues. In addition, an interesting phenomenon is ob-served. Due to a stochastic nature of the secondarydendrite arm formation process, one of the secondaryarms of the primary dendrite stalk aligned with the x-axisis growing at a appreciable speed to block the growth ofsecondary arms associated with the primary stalk alignedwith the y-axis. When the same simulation was repeatedwith a different set of seed numbers, (used by the randomnumber generator during computation of the noise term),the same phenomena was not observed and secondaryarms were developed to comparable extents on both pri-mary dendrite stalks.

Figure 3(d) shows that between 2.25 ms and 3.0 msthe same growth pattern of the dendrite, established atshorter simulation times, is retained. The maximumtemperature increase at the solid/liquid interface during


Fig. 3. Contour plots of the mole fraction of copper at four different simulation times: (a) 0.75 ms, (b) 1.5 ms,(c) 2.25 ms, and (d) 3.0 ms during isothermal solidification of a Ni-0.41Cu alloy at 1574 K.

isothermal solidification of the Ni-0.41Cu alloy from asupersaturated liquid at 1574 Kis found to be ,1.7 K.

3.2. Case B: Solidification under Non-isothermalConditions

The phase-field model developed in Section 2.1 isused in the present section to study the development of thedendrite microstructure in the Ni-0.41Cu alloy under threedifferent externally imposed cooling rates, 2.0 3 103 K/s,2.0 3 104K/s, and 1.0 3 105 K/s. The temperature evolu-tion equation, Eq. (24) is considered in this case and amuch smaller time increment (based on the heat conduc-tion Courant number d2 c/K 5 7.0 3 10211 s) is used.

Figures 4(a)-(d) show the dendrite microstructureand the corresponding mole fraction of copper and tem-

perature fields at two different simulation times (1.5 msand 3.0 ms) during non-isothermal solidification of theNi-0.41Cu alloy under the lowest externally imposedcooling rate of 2.0 3 103 K/s. It should be noted that thesame scale is used for the mole fraction of copper in Fig-ures 4(a) and (c) (as well as in all the subsequent figuresdisplaying the mole fraction of copper contour plots) asin the isothermal solidification case, Figures 3(a)-(d). Asfar as the temperature scale is concerned, to help revealrelationships between the morphology of the solid/liquidinterface and the surrounding temperature field, only a0.06 Ktemperature range (characteristic for the solid/liq-uid interfacial region) is shown in Figures 4(b) and (d)(as well as in all the subsequent figures displaying thetemperature contour plots). To provide additional infor-mation about the temperature field, the maximum and theminimum temperatures of the computational domain are


Fig. 4. Contour plots of the mole fraction of copper, (a) and (c), and contour plots of the temperature, (b) and(d) at two different simulation times: (a) and (b) 1.5 ms and (c) and (d) 3.0 ms during solidification of a Ni-0.41Cu alloy under the externally imposed cooling rate of 2.0 3 103 K/s.

also included in Figures 4(b) and (d), as well as in all theother figures displaying the temperature contour plots.

Figure 4(a) shows that at the shorter simulation time(1.5 ms), perturbations of the solid/liquid interface areclearly visible but they do not result in interface insta-bility. This is consistent with the fact that, as shown inFigure 4(b), the temperatures of the liquid in contact withthe solid at this simulation time are higher than the tem-perature used in the isothermal solidification case(1574 K) and hence the liquid is less undercooled. Acareful observation of the Figure 4(b) shows a phenom-enon which is frequently observed. That is the highesttemperatures are generally associated with a pocket ofliquid which is in the processes of being cut off from therest of the liquid. As solidification takes place in suchpockets of the liquid, the latent heat of fusion has to beextracted through the surrounding solid which, in turn,becomes subjected to the higher temperatures. When thesolidification of such liquid pockets is near completion,and less latent heat of fusion is being released, the tem-perature starts to decrease. Simultaneously, pockets ofisolated liquid are formed at other locations along thesolid/liquid interface. Consequently, the maximum tem-perature in the computational domain evolves during so-lidification both temporally and spatially.

The lowest temperatures at the solid/liquid interfaceare typically found at the (high curvature) tips of the pri-mary dendrite stalks [e.g., Figure 4(b)]. This finding isconsistent with the Gibbs-Thomson effect which statesthat, due to a curvature-induced increase in the free en-ergy of the solid, a larger undercooling of the melt is re-quired to obtain the same thermodynamic driving forcefor solidification.

Figures 4(c) and (d) show that under the same ex-ternally imposed cooling rate of 2.0 3 103 K/s but at alonger simulation time (3.0 ms), secondary dendritearms have formed but their extension into the under-cooled liquid is quite limited. This finding is fully con-sistent with the fact that the liquid is less undercooledunder these solidification conditions. How dramatic canbe the effect of liquid undercooling on the dendrite mi-crostructure can be seen by comparing the dendrite mi-crostructures obtained under the isothermal solidifica-tion conditions, Figures 3(b) and (d), with theircounterparts obtained during solidification under the ex-ternally imposed cooling rate of 2.0 3 103 K/s, Figures4(b) and (d). In the former case, secondary dendrite armsare fully developed and there are even signs that tertiarydendrite arms are beginning to form. In the latter case,on the other hand, secondary dendrite arms are not welldeveloped and the solid particle has essentially a four-pointed star shape.

Examination of the results pertaining to the distribu-tion of the copper mole fraction throughout the interfacialregion (the results not shown here for brevity) indicate thatthe mole fraction of copper at the solid/liquid interface(defined as f 5 0.5) is consistently lower than its coun-terpart in the isothermal solidification case. This finding isconsistent with the fact since the interfacial temperaturesin the non-isothermal case are generally increased relativeto the ones in the isothermal case (T 5 1574 K), the equi-librium mole fractions of copper in both the solid and theliquid are lower (please consult Figure 2).

Figures 5(a)-(d) show the dendrite microstructure andthe corresponding mole fraction of copper and the temper-ature fields at two different simulation times (1.5 ms and3.0 ms) during non-isothermal solidification under theexternally imposed cooling rate of 1.0 3 104 K/s.

At the shorter of the two simulation times (1.5 ms),Figures 5(a) and (b), the solid particle acquires a dia-mond like shape with a number of not well developedsecondary arms on its surface. Due to a higher rate ofheat extraction, the temperatures in the vicinity of thesolid/liquid interface is quite near the initial temperature1574Kand, hence, undercooling of the liquid is compa-rable to that in the case of the isothermal solidification.Furthermore, a comparison of the results displayed inFigures 4(a) and 5(a) shows that an increase in the heatextraction rate gives rise to a decrease in the diffusionlengths, both in the liquid and in the solid.

Figures 5(c) and (d) show that during subsequentsolidification under the same externally imposed coolingrate of 2.0 3 104 K/s, few secondary dendrite arms beginto advance into the surrounding undercooled liquid. Thelatent heat of fusion released during solidification in-creases the temperature (particularly in the solid) but thisincrease in less than 50% of that observed during solidi-fication under the lowest cooling rate used. Conse-quently, the liquid near to interface is sufficiently under-cooled to promote the development of the dendritemicrostructure.

Figures 6(a)-(d) show the dendrite microstructure andthe corresponding mole fraction of copper and the temper-ature fields at two different simulation times (1.0 ms and1.8 ms) during non-isothermal solidification under thehighest externally imposed cooling rate of 1.0 3 104 K/s.It should be noted that dendrite microstructure at shortersimulation times (1.0 ms and 1.8ms) are displayed in thesefigures relative to the ones in Figures 4(a)-(d) and 5(a)-(d)because, due to a significantly larger growth rates in thepresent case, the primary dendrite stalks tend to grow outof the computational domain at longer simulation times.

At the shorter of the two simulation times (1.0 ms),Figures 6(a) and (b), the solid particle has a curved


diamond shape and its surface is quite smooth, except nearthe dendrite tips where single cells are formed. No evi-dence of cell formation is observed at a somewhat shortersimulation time (0.9 ms) but these results are not shownhere for brevity. The copper mole fraction contour plotdisplayed in Figure 6(a) shows that partitioning of the cop-per between the solid and the liquid is quite limited andthat the solute (copper) is being trapped by the fast ad-vancing solid/liquid interface. Figure 6(b) shows that dueto a high heat extraction rate, temperatures around thesolid/liquid interface are lower by ,3 K than the initialtemperature. A larger liquid undercooling would generallypromote solid/liquid interface instability and formation ofsecondary dendrite arms. However, due to a high rate ofheat extraction, the interface is advancing at a high speedinto the undercooled liquid and most of the interfacialfluctuations do not give rise to interface instability.

Figures 6(c) and (d) show that during subsequentsolidification under the same highest externally appliedcooling rate of 1.0 3 105 K/s, the solid particle has re-tained its curved diamond shape. However, the solid/liq-uid interface is not smooth any longer and contains anumber of “cells”. The interface perturbations are nowbeing referred to as cells and not as secondary dendritearms because they tend to grow in a direction of the localinterface normal and not orthogonal to the direction ofthe primary stalks from which they originate. For exam-ple, the cells adjacent to the primary stalks in Figures6(c) and (d) grow in a direction parallel to the adjacentprimary stalk rather than in a direction normal to it. Asimple analysis of the results displayed in Figure 6(c)shows that solute trapping continues to take place be-tween 1.0 ms and 1.5 ms. Figure 6(d) indicates that, dueto release of the latent heat of fusion, temperatures


Fig. 5. Contour plots of the mole fraction of copper, (a) and (c), and contour plots of the temperature, (b)and (d) at two different simulation times: (a) and (b) 1.5 ms and (c) and (d) 3.0 ms during solidification ofa Ni-0.41Cu alloy under the externally imposed cooling rate of 2.0 3 104 K/s.

around the solid/liquid interface have increased some-what relative to the initial temperature. However, a com-parison of the results shown in Figure 6(d) with thoseshown in Figure 4(d) and 5(d) indicates that liquid un-dercooling is considerably larger for the solidificationcase under the highest externally imposed cooling rate.As discussed earlier, while a higher liquid undercoolingpromotes interface instability, interfacial perturbationscan not break away from the interface into the unde-cooled liquid due to a large speed of the solid/liquid in-terface. The net result is a solid/liquid interface contain-ing a variety of cells which extend deep into the bulk ofthe solid particle.

The solute trapping phenomenon was previouslystudied by Ahmad et al. [11] who showed that for a planarsolid/liquid interface occurs when the solute diffusionlength ahead of the interface (,D/v1, where v1 is the inter-

face velocity) becomes comparable with the interfacethickness, d. If the solute diffusivity at the interface is setto D(f 5 0.5) 5 5.0 3 10210 m2/s, the critical velocityneeded for solid trapping is ,0.01 m/s. Using the lengthof the primary dendrite stalks and the simulation time inFigure 7(c) and (d), the average growth velocity of the tipsof the primary dendrite stalks is assessed as 0.005 m/s.This simple analysis confirms that under the highest exter-nally imposed cooling rate of 1.0 3 105 K/s, the conditionsare meet (or are nearly being meet) for solute trapping.

The evolutions of the maximum (solid lines) and theminimum (dashed lines) temperatures in the computa-tional domain with the simulation time for the three dif-ferent cooling rates (2.0 3 103 K/s, 2.0 3 104 K/s and1.0 3 105 K/s) are shown in Figure 7. The results dis-played in this figure show that, at the intermediate (2.03104 K/s) and the highest (1.0 3 105 K/s) cooling rates,


Fig. 6. Contour plots of the mole fraction of copper, (a) and (c), and contour plots of the temperature, (b) and(d) at two different simulation times: (a) and (b) 1.0 ms and (c) and (d) 1.8 ms during solidification of a Ni-0.41Cu alloy under the externally imposed cooling rate of 1.0 3 105 K/s.

the system temperature initially decreases. This can beexplained by the fact that partitioning of the solute be-tween the solid and the liquid and its diffusion are lim-ited and, hence, larger liquid undercooling is needed toachieve the same level of thermodynamic driving forcefor solidification. However, once the solid particle starts

to grow, the latent heat of fusion is released giving riseto an increase in the temperature, i.e. to the recalesence.To sharp contrast, under the lowest (2.0 3 103 K/s) cool-ing rate imposed, solute portioning and diffusion readilytake place and a lower level of liquid undercooling isneeded for the solidification to proceed. Consequently,the release of the latent heat of fusion is sufficiently fastfrom the beginning of the simulation, giving rise to acontinuous rise in the temperature of the system.

A comparison of the solid and dashed lines corre-sponding to the same imposed cooling rate in Figure 7indicates that particularly at the highest cooling rate im-posed, the spatial variation in the temperature (measuredby the difference between the maximum and the mini-mum temperatures) can be significant (as high as 6 K).Hence, the approximation about spatial uniformity of thetemperature frequently employed to simply simulationsof the solidification process under non-isothermal condi-tions appear to be justified only under relatively low(,2.0 3 103 K/s) cooling rate.

4. CONCLUSIONS

Based on the results obtained in the present work,the following main conclusions can be drawn:

1. Spatial nonuniformities in the temperature dur-ing solidification can be neglected only under therelatively low cooling rates (,2.0 3 103 K/s). At


Fig. 7. The evolution of the maximum (solid lines) and the minimum(dashed lines) temperatures in the computational domain as a functionof the simulation time for three values of the externally imposedcooling rate.

Table I. Material Parameters for Pure Nickel and Copper

ElementEquation where

Parameter Symbol Units Ni Cu first used

Melting point Tm K 1728 1358 (2)Latent heat of fusion L J/m3 2350 3 106 1728 3 106 (9)Molar volume Vm m3 7.42 3 1026 7.42 3 106 (4)Surface energy j J/m2 0.37 0.29 (2)Interface kinetic coefficient b m/K/s 3.3 3 1023 3.9 3 1023 (14)Diffusion coefficient in liquid DL m2/s 1029 1029 (20)Diffusion coefficient in solid Ds m2/s 10213 10213 (20)Interface thickness d m 5.0 3 1028 5.0 3 1028 (2)

Table II. Phase Field Parameter for the Ni-Cu Alloy System

Parameter Symbol Units Value Equation where first used

Surface energy anisotropy amplitude g N/A 0.04 (2)Interface mode number k N/A 4 (2)Noise amplitude a N/A 0.3 (13)Regular solution parameter in liquid VAB,L kJ/mole 8.5 (12)Regular solution parameter in solid VAB,S kJ/mole 11.2 (12)

higher cooling rates, spatial non-uniformity intemperature can be significant (e.g. as high as 6 Kat a cooling rate of 1.0 3 105 K/s).

2. At lower cooling rates, solidification is con-trolled by solute partitioning (diffusion) while athigh cooling rate solute trapping takes place andsolidification is controlled by heat extraction.

3. Cooling rate has a pronounced effect on the mor-phology of the dendrites causing it to vary, as thecooling rate increases, from undeveloped den-drites consisting only of primary stalks, via fullydeveloped dendrites containing secondary andtertiary arms, to the cellular microstructure.

ACKNOWLEDGMENTS

The material presented in this paper is based on worksupported by the U.S. Army Grant Number DAAD19-01-

1-0661. The authors are indebted to Drs. Walter Roy,Fred Stanton, Bonnie Gersten, and William DeRossetof ARL for the support and a continuing interest in thepresent work.

REFERENCES

1. J. A. Warren and W. J. Boettinger, Acta Metal.43, 689 (1995).2. J. Z. Zhu, Z. K. Liu, V. Vaithyanathan, and L. Q. Chen, Script

Mater. 46, 401 (2002).3. G. Caginalp and J. Jones, Ann. Phys.237,66 (1995).4. G. Caginalp and W. Xie, Phys. Rev. A48, 1897 (1993).5. R. Kobayashi, Acta Metal.43, 689 (1995).6. J. A. Warren and B. T. Murray, Model. Simulation Mater. Sci. Eng.

4, 215 (1996).7. J. A. Warren, IEEE Comput. Sci. Eng.2, 38 (1995).8. W. J. Boettinger and J. A. Warren, Metall. Mater. Trans.27A, 657

(1996).9. I. Loginova, G. Amberg, and J. Agren, Acta Mater.49,573 (2001).

10. A. A. Wheeler, W. J. Boettinger, and G. B. McFadden, Phys. Rev.A 45, 7424 (1992).

11. N. A. Ahmad, A. A. Wheeler, W. J. Boettinger, and G. B.McFadden, Phys. Rev. E58, 3436 (1998).


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