velocity selection in 3ddendrites: phase ﬁeld computations and microgravity...

Nonlinear Analysis 62 (2005) 467–481www.elsevier.com/locate/na

Velocity selection in 3D dendrites: Phase fieldcomputations and microgravity experiments

Y.B. Altundasa,∗, G. CaginalpbaInstitute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA

bDepartment of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Received 18 February 2004; accepted 21 February 2005

Abstract

The growth of a single needle of succinonitrile (SCN) is studied in three-dimensional (3D) spaceby using a phase field model. For realistic physical parameters, namely, the large differences inthe length scales, i.e., the capillarity length (10−8–10−6 cm), the radius of the curvature at the tipof the interface (10−3–10−2 cm) and the diffusion length (10−3–10−1 cm), resolution of the largedifferences in length scale necessitates a 5003 grid on the supercomputer. The parameters, initial andboundary conditions used are identical to those of the microgravity experiments of Glicksman et al.for SCN. The numerical results for the tip velocity are (i) largely consistent with the Space Shuttleexperiments, (ii) compatible with the experimental conclusion that tip velocity does not increase withincreased anisotropy, (iii) different for 2D versus 3D by a factor of approximately 1.9, (iv) essentiallyidentical for fully versus rotationally symmetric 3D.� 2005 Elsevier Ltd. All rights reserved.

PACS:68Q10; 82C24; 74N05; 35K50

Keywords:Dendritic growth; Phase field equations; Parallel computing; Microgravity experiments; 3Dsolidification calculation

∗ Corresponding author.E-mail addresses:[email protected](Y.B. Altundas),[email protected](G. Caginalp)URL: http://www.pitt.edu/∼caginalp(G. Caginalp).

0362-546X/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2005.02.122

http://www.elsevier.com/locate/na

mailto:[email protected]

mailto:[email protected]

http://www.pitt.edu/~caginalp

468 Y.B. Altundas, G. Caginalp / Nonlinear Analysis 62 (2005) 467–481

1. Introduction

The temporal evolution of an interface during solidification has been under intensivestudy by physicists and material scientists for several decades. The interface velocity andshape have important consequences for practical metallurgy, as well as the theory, e.g.,velocity selection mechanism and nonlinear theory of interfaces.The simplest observed microstructure is the single needle crystal or dendrite, which is

observed to be a shape resembling a paraboloid (but not fully rotationally invariant awayfrom the tip) growing at a constant velocity,v0, with tip radius,R0.An early model of this phenomenon by Ivantsov[19] stipulated the heat diffusion equa-

tion in one of the phases and imposed latent heat considerations at the interface. With theinterface assumed to be at themelting temperature, the absence of an additional length scaleimplies the existence of an infinite spectrum of pairs of velocities and tip radii,(v0, R0).Since the experimental results have shown that there is a unique pair(v0, R0) that is in-dependent of initial conditions, there has been considerable activity toward uncovering thetheoretical mechanism for this velocity selection (see, for example[5,6,21,27]). The emer-gence of the capillarity length associated with the surface tension as an additional lengthscale has provided an explanation for the selection mechanism.Advances in computationalpower and a better understanding of interface models and their computation have openedup the possibility of comparing experimental values for the tip velocity with the numericalcomputations. This is nevertheless a difficult computational issue in part due to the largedifferences in length scales that range from 1cm for the size of the experimental region, to14�m for the radius of curvature near the tip, 10−6 cm for the capillarity length, to 10−8 cminterface thickness length.One perspective into the theoretical and numerical study of such interfaces has been

provided by the phase field model introduced in[7,8] in which a phase, or order parameter,�, and temperature,T, are coupled through a pair of partial differential equations describedbelow (see also more recent papers[2,18,23]). In physical terms, the width of the transi-tion region exhibited by� is Angstroms. In the 1980s three key results facilitated the useof these equations for computation of physical phenomena. If the equations are properlyscaled one can (i) identify each of the physical parameters, such as the surface tension, (ii)and attain the sharp interface problem as a limit[9], and (iii) use the interface thickness,�, as a free parameter, since the motion of the interface is independent of this parameter[12]. This last result thereby opened the door to computations with realistic material pa-rameters, by removing the issue of small interface thickness. However, the difference inscale between the radius of the curvature and overall dimensions still pose a computationalchallenge. More recently, several computations, have been done using the phase fieldmodel[1,20,28,29,31,32], with some three-dimensional (3D) computations in[20] utilizing themodel and asymptotics of[11], that will be compared with our results below. Also, GeorgeandWarren studied[14] the simulation of dendritic growth in 3D space using a phase fieldmodel.Our work differs from the works referenced above in many aspects. However, the main

difference arises from the adaptation of the experimental conditions in the simulation ofdendritic growth. Most importantly, we use true values of physical parameters which areobtained from the microgravity experiment for SCN[24]. In order to deal with different

Y.B. Altundas, G. Caginalp / Nonlinear Analysis 62 (2005) 467–481 469

length scales and the diffusion during freezing in the thin interfacial region, we implementa fully parallel architecture in a 3D space which enables us to use enough grid points andperform an efficient calculation.Solidification is a complicated nonlinear process. Modeling necessarily involves making

choices of physical effects that are to be included in the equations. Comparison of computa-tionswith experiments that are closest to themathematicalmodel yields themost convincingtest of the model and computations. The modeling of single-needle dendrites has usuallybeen carried out using the diffusion equation as a mechanism for the dissipation of heat.However, all of the experiments until the Space Shuttle experiments had been done underconditions of normal gravity, so that convection in the liquid is an important mechanismfor the dissipation of the latent heat released at the interface. The microgravity experimentsperformed on the Space Shuttle[24] provide the first opportunity to test whether the mathe-matical models agree with experiments, since the absence of gravity essentially eliminatesconvection, thereby leaving diffusion as the main mechanism for heat transport away fromthe interface.While numerous computer calculations have been performed on both sharp interface and

phasefieldmodelsof solidification, comparisonwithexperiment hasalwaysbeenadifficultydue to the vastly different length scales in the problem (e.g 10−6 cm for capillarity lengthand 1 cm for the overall dimensions of the experiment), and the 3D nature of the problem.In the absence of direct comparison with experiment, it is also difficult to know whethersome of the simplifications that have been used, such as setting the kinetic coefficient,�, tozero are valid.In this paper we perform large scale 3D parallel computations of a phase fieldmodel with

the modification introduced in[11]. The key aspects of these computations are summarizedbelow.(A)We perform fully 3D parallel computations by adopting the experimental conditions

used in the Space Shuttle experiment. The symmetry is utilized only along the major axes(rather than rotational symmetry). This allows us to compare the tip velocity with the actualexperiments in ameaningful way. The calculations utilize the parameters and boundary con-ditions of the IDGE microgravity experiments for SCN[16,24]. All previous experimentsdone under normal gravity conditions introduced convection. Hence this provides an op-portunity to compare experiments in the absence of convection to theory that also excludesconvection. The difference between the experimental results and our computations therebydefines the challenges for additional physical effects that need to be modeled.(B) The role of anisotropy in velocity selection has been noted in the computational ref-

erences cited above. Glicksman and Singh[17] compare experimental tip velocity of SCNwith pivalic acid (PVA) whose coefficient of surface tension anisotropy (defined below) dif-fers by a factor of 10 but are otherwise similar, except perhaps for the kinetic coefficient.Weperform two sets of calculations in which all parameters are identical (SCN values) exceptfor the anisotropy coefficient. Our computations confirm (consistent with the experimentalresults[24]) that the velocity is nearly identical when the magnitude of the anisotropy isvaried by a factor of 10 with all other parameters fixed (at the SCN values).(C) Most of the previous numerical computations that simulate the interface growth

were done in two-dimensional (2D) space. Our computations shows that the 2D and 3Dcomputations differ by a factor of approximately 1.9. The results of the 3D for tip velocity


can also be compared with our previous computations[3] that utilized rotational symmetryto reduce the 3D computations to two computational spatial dimensions.(D) The role of the kinetic coefficient (see definition of� below Eq. (2)) is subtle, and this

material parameter is often set to zero, for convenience, in theoretical and computationalstudies. We find, however, that there is a significant difference in the tip velocity when allother parameters are held fixed while this coefficient is varied. Consequently, this kineticcoefficient may be of crucial importance in determining the selection of tip velocity. Abetter understanding of this issue may lead to theory that can explain a broader range ofundercooling and velocity.

2. Mathematical modeling

In the computations below, we use a version of the phase field equations introduced in[11], for which the phase or order parameter,�(�x, t), as a function of spacial point,�x, andtime, t, is exactly−1 in the solid and+1 in the liquid. The order parameter is coupled withthe dimensionless temperature,u, which is given by the following relation along with thecapillary length,d0.

u(x, t) = T − Tm

lv/cv, d0 = �cv

[s]Elv, (1)

whereTm, lv, cv, � and[s]E are the melting temperature, latent heat, specific heat per unitvolume of the material, surface tension and the difference in the entropy (in equilibrium)per unit volume between the solid phase and liquid phase, respectively. Thus, we can definethe interface by�={x ∈ � : �(x, t)=0} and write the dimensionless phase field equationsas follows:

��2�t = �2�� + g(�) + 5

8

�d0

uf ′(�), (2)

ut + 1

2�t = D�u, (3)

where

g(�) = � − �3

2, f ′(�) = (1− �2)2, D = K

cv. (4)

Here,� is the kinetic coefficient and� is the interface thickness that can be used as a “freeparameter”[12]. In the limit as� vanishes as all other parameters held fixed, solutions to(2) and (3) are governed by the sharp interface model

ut = ∇ · D∇u, (5)

vn = −D[∇u · n̂]+−, (6)

u = −d0(� + �vn), (7)

where the parametersd0, D and� are the same as in the phase field model, andvn is theinterface normal growth velocity (with normaln̂ chosen from solid (−) to liquid (+)) [8,11].


X

Y

0 50 1000

102030405060708090

X

Y

0 50 1000

102030405060708090

Fig. 1. Contour plots of the interface at different times onxyplane foru∞ = 0.0265 which shows the effect of theanisotropy: (a) The position of the interface in 10 s; (b) The position of the interface at latter times (135, 140, 145and 150 s).

2.1. Initial and boundary conditions

In order to simulate interface growth of a dendrite in 3D, we choose a cube of[−1,1]3which is assumed to be filled with pure SCN melt initially. The solidification of the meltis initiated by a small solid SCN ball of radius,R0, which is placed at the center of thechamber. The temperature at the boundary is kept at constant undercooling value,u∞, andthe liquid temperature inside the chamber declines exponentially fromu = usolid on theinterface of the seed to the boundary of the chamber. In particular, the initial conditions ofu inside the chamber are given by a plane wave solution to (5) and (6) which is given by

utrav(z, t) ={u∞[1− e−v(z−vt/|u∞|)/(D|u∞|)], z�vt/|u∞|,0, z< vt/|u∞|, (8)

wherez is the signed distance from the seed interface (positive in the liquid) andu∞ =(T∞ −Tm)/(lv/cv) denotes the dimensionless undercooling value whereT∞ is the far fieldtemperature. The initial value of� is obtained from a leading term asymptotic expansionsolution[11]

�(x, t) = tanh

(z − vt

2�

)+ higher order terms. (9)

2.2. Implementation of anisotropy

Anisotropy is important in determining the shape of dendrites that grow exclusively in thepreferred directions. The experimental evidence shows conclusively that surface tension,�, exhibits anisotropy,[17]. While there is the possibility of dynamical (i.e., through� inEqs. (2) or (3)) or other anisotropy the experimental measurements of anisotropy in theseexperiments are confined to those related to surface tension. Surface tension anisotropy hasbeen modeled in several ways (Fig.1).


Let n̂ = (nx, ny, nz) be the normal to the interface. We utilize the simplest possiblefunction describing the dependence of surface energy onn̂ in the case of an underlyingcubic symmetry. The relation can be given by[22,33]

�(n̂) = as[1+ �(n4x + n4y + n4z)] (10)

which is rewritten in terms of spherical angles as

�(,) = as{1+ �[cos4 + sin4(1− 2 sin2 cos2 )]}, (11)

where and are the angles which correspond to the normal,n, with respect to a crystalaxis. The parameters,� andas , can be related to usual measure of anisotropy strength,��,by the relationsas = (1− 3��) and� = 4��/as [20].Asymptotic analysis[10] shows that with the anisotropy, the Gibbs–Thomson relation

(7) is modified not only in terms of the angles, but with their second derivatives. In oursimulation, we assume that the dendrites grow along an axis of symmetry and set� to bethe mean curvature. Thus, we can rewrite the Gibbs–Thomson Eq. (7) as

u = −d0(n̂)� − d0(n̂)�vn, (12)

where

d0(,) = d0

[�(,) + �2�(,)

�2+ �2�(,)

�2

]. (13)

3. Discretization

A large number of mesh points are necessary in order to calculate the tip growth velocityand tip radius accurately. The values ofu and� across the interface vary from maximumto minimum within a short distance. This requires finer grid spacing so that number ofmesh points is adequate to resolve the interfacial area.Wemake a physically reasonable butcomputationally very useful assumption that the dendrite grows symmetrically along thecoordinate axes. Thus, the computational domain is reduced to�=[0,1]3 which decreasesthe overall grid usage by 7/8. In this work, we use 400–500 uniform grid points on eachside of� which guarantees that we have at least 6–7 grid points located on the interfaceregion as measured from� = −0.9 to� = 0.9.In order to discretize Eqs. (2) and (3), let�p

ijk andupijk denote the discrete values of

�(xi, yj , zk, tp) andu(xi, yj , zk, tp), respectively. We lag the nonlinear terms in (2) anddiscretize Eqs. (2) and (3) by using semi-implicit Crank–Nicolson finite difference method.Thus, we write the discrete equations as follows:

�+t u

pijk = D

2{�u

pijk + �u

p+1ijk } − 1

2�+t �p

ijk, (14)

�2��+t �p

ijk = �2

2{��p

ijk + ��p+1ijk } + g(�p

ijk) + 5

8

�d0

upijkf

′(�pijk), (15)


where

�Tijk = �±

xiT

ijk + �±yj

Tijk + �±

zkT

ijk (16)

for i, j, k =0,1,2, . . . , N −1 andp =0,1,2, . . . , Tfinal. The operators such as�±x and�

+t

in Eqs. (14) and (15) are the finite difference operators and they can be given as

�±x (x, y, z, t) = (x + �x, y, z, t) + (x − �x, y, z, t) − 2(x, y, z, t)

(�x)2(17)

and

�+t (x, y, z, t) = (x, y, z, t + �t) − (x, y, z, t)

�t· (18)

4. Values of SCN parameters

The comparison of the computational results with experiments makes sense only if weuse the true value of the physical parameters. Throughout the simulation, the parametersd0 and�0 for SCN are set to be 2.83× 10−7 cm and 8.9ergs/cm2, respectively[16]. Thediffusivity parameters,Dliquid andDsolid, are almost the same for the liquid and solid SCN.Thus, we set the diffusivityD to be 1.147× 10−3 cm2/s which is the value forDliquid [16].All of the parameters in the phase field equations are physically measurable quantities,

including � which is a measure of the interface thickness. It was shown in[8,9] that thesolutions of the phase field equations (scaled in this form) approach those of the sharpinterface model (5)–(7) provided all other parameters are held fixed as� approaches zero.The rate of convergence, however, emerges as a key issue. In particular, the true size of� isa few atomic lengths, or Angstrom, while the size of experimental region is at least 1 cm.Thus using the true value of� would necessitate 109 grid points in each direction yieldingunfeasible computation. Fromacomputational perspective, oneneeds tohaveat least severalgrid points in the interfacial region in order to accurately calculate the derivatives of orderparameter that implicitly define the surface tension.A computational breakthrough was the discovery that� could be made many orders of

magnitude largerwithout influencing the interfacemotion.CaginalpandSocolovsky[12,13]showed that so long as one chooses an appropriate number of grid points in the interfaceregion (defined by the magnitude of� andh denotes the uniform grid size), guaranteed bythe range 0.75< �/h<1.1, one can resolve the motion accurately. The only limitation theninvolves the interface thickness relative to the radius of curvature of the dendrites. In thiswork we set� = h which falls into this range.The choice of the initial tip radius,R0, for a steady-state is not arbitrary. One needs to

take into account the latent heat released at the interface. By choosing a sufficiently largedistance between the interface and the boundary, the latent heat released at the interfacediffuses to the liquid and the effect of the boundary becomesminimal. In order to guaranteeenough distance to the boundary, the tip radius,R0, should be at least 20 times smallerthan the diffusion lengthD/vn. In our calculation, we choose the tip radius to beR0= 20hfor the choice ofN = 500 andR0 = 14h for N = 400 whereh is the uniform grid size


corresponding to the choice of eachN. Under these conditions, the diffusion length is largeenough and satisfies the standard theoretical conditions for dendritic growth[28].

5. Parallelization and data distribution

Thenumerical simulation of Eqs. (2) and (3) in 3Dwith any physical choice of parametersis a difficult task. Of these difficulties, the memory requirements and the CPU time are themain issues due to the difference in the length scales. As shown inTable 1, the demandfor the memory is a delicate issue in that doubling the number of the grids will increasethe computational memory as much as eight times, and slows down the performance of thecode.Thismakes thenumerical computationof (14)and (15)withanyphysically appropriatechoice of grid size impossible on serial computers. Instead, one needs a parallel architecturein which the work will be distributed to many computers and the computational job will beshared among the computers (processors) allowing large scale computation. In this work,we use PETSC’s distributed memory architecture (DA), whose characteristic feature is thateach processor owns its own local memory, and memory of other processors cannot beaccessed directly[4]. The PETSC/DA system requires communication to inquire or borrowinformation among the processors. In particular for the numerical solution of PDE’s, eachprocessor requires its local portionof the informationaswell as thepoints on theboundary ofthe adjacent processors to update the right-hand side vector. The communication requiredamong the processors to exchange the components and points along the border of theadjacent subdomains are managed via the DA system while the actual data is stored inappropriately sized local vector objects. Thus, the DA objects only contain the parallellayout and communication information, and they are not intended for storing the matricesand vectors.The communication is necessary but it is very critical that the parallel code be designed

independent of the other processors as much as possible and the ratio of communicationamong the processors should be kept small. Otherwise, a high ratio of communicationsamong the processors slows down the computation. Therefore, it is very important that thecommunication should be limited to the neighboring processors and should avoid globalcommunication if possible. Similarly, the distribution of thework loadamong theprocessorsis another important issue in parallel computation. In our work, we keep the number of gridpoints proportional to the number of processor so that each processor is assigned almostthe same amount of work load. This enables the efficient use of the processors and makes

Table 1Table shows the memory allocation on each processor (of 32 processors) when the number of grid points aredoubled

Grid points Memory (MB) Ratio

323 1.45 —643 10.56 7.281283 76.90 7.282163 611.03 7.95


Table 2The performance of the algorithm is tested for fixed grid (N = 128): Table shows the wall-clock time, number offlops and the rate of scalability of the algorithm on different number of processors

Number of PE MFlops/s Memory (MB) Wall Clock (s) Ratio

4 50 492.8 1165.78 —8 52 255.5 574.58 2.0016 54 136.5 314.15 3.7132 55 76.9 168.87 6.9064 53 47.0 88.59 13.15

the processors to work in a synchrony.Table 2shows that both the CPU time and memoryallocation are almost halved when the number of the processors doubled.The allocation of memory, creation of the parallel matrices and vectors, and setting up the

solver contexts are very time consuming. Therefore, one would like to use the same initialsetup throughout the computation if possible. This is a good approach especially when thecoefficientmatrix is independent of timewhich is the case in this work. Thus, we can use thesame coefficient matrices as well as the same preconditioners throughout the calculations.Parallel solutions of (14) and (15) are done via the linear solver of PETSC[4] in which

we use CG iterative method with Jacobi Preconditioning.

5.1. Memory allocation and scalability of the algorithm

In the numerical solution of (14) and (15), we allocate memory for six parallel andone sequential global vectors of the sizeN3, and two parallel global matrices of the sizeN3 × N3. Together with the creation of the DA system, local vectors and matrices, thememory requirement becomes huge for largerN. We verify this inTable 1by varying thenumber of grid points. In fact, it shows that asN is doubled, thememory on each processor isincreased approximately eight times. Thus asN increases, the demand for the memory getsso large that even for supercomputers such as Lemieux, there are considerable limitationsof the number when grid points are larger thanN = 600. AsTable 1indicates, the memoryrequired for a fully 3D computation of phase field model is enormous if one wants touse a reasonable number of grid points in the simulation. This necessitates the use of notonly high-performance computers but also computers which can accommodate thememoryneeded. One way to over come this difficulty is to use parallel architecture which is the keyin our work in handling the memory deficiency. The scalability of the code is a measureby which one can test whether the processors are efficiently used during the computation.For this we fix the number of the grid points atN = 128 and setTfinal = 300. By doublingthe number of the processors each time we calculate the corresponding wall-clock time forthe same job. As shown inTable 2the wall-clock time almost doubles when we halved thenumber of the processors. This is an indication that the code scales well with the numberof the processors.As it is apparent from above analysis, memory allocation is still a delicate issue which

greatly influences the choice of grid sizeN. In our earlier work[3], we studied the grid


Table 3A sufficient grid size for accurate calculation of interphase growth velocities for SCN is examined

Grid number Velocity (cm/s)

200 0.002100300 0.000363400 0.000310500 0.000327600 0.000342700 0.000357

The velocities foru∞ = 0.01 are shown attfinal = 16 s from the initial stage

convergence in 2D by comparing the growth velocities for different choice of grid pointsranging from200 to 700whenall other parameters kept fixed.Table 3shows the correspond-ing velocities for the grid points from 200 to 700 when all other conditions are identicalfor the undercooling value 0.001. As seen inTable 3, the interface growth velocity doesnot differ much when we vary the number of grid points,N, from 400 to 700 indicatingthat a choice ofN = 500 will be enough to study dendritic growth. In this work, we useN = 500 in the calculation of undercooling versus growth velocity linearity relation. Forother calculations such as the study of anisotropy and kinetic coefficient, we setN = 400.Also, the time step�t is chosen to be 5× 10−3 throughout the computation.

6. Results and conclusions

In this paper we have performed parallel computations in 3D space with the specificparameters and boundary conditions which are used in the microgravity experimental setupfor SCN (Fig.2).The microgravity experiments exclude most of the convection effects so that compu-

tations involving the physics described by Eqs. (2)–(3) or (5)–(7) can be tested againstthe experiment. Prior to these experiments, theoretical results and computer calculationswere awkward in that theory without convection was tested against experiment (on Earth)with convection. Thus the interpretation of agreement was ambiguous, leaving open thepossibility of inaccurate computations on inadequate modeling of experimental setup.In order to address the questions raised in (A) and (B) of the introduction, we have

considered eight different undercooling values,u∞, from the microgravity experiments forSCN[15]. During the simulationwe setR0=20h,N=500 andTfinal=2000 and compute theaverage growth velocity for each undercooling value. The computational and experimentalgrowth velocities for each undercooling are given inTable 4.The results inTable 4and corresponding Fig.3 show that computational velocity is

consistent with other 3D phase field computations (see e.g.[3]). The overall results areclose to the experiments, particularly for undercooling temperatures that are neither verysmall nor very large. In particular, since solidification is a complicated process, it is likelythat many other physical effects play a role in determining the growth velocity at the tipof the dendrite[30]. Eqs. (5)–(7) or equivalently (2) and (3) incorporate all of the physics


X

Y

0 50 1000

102030405060708090

100110

0

20

40

60

80

100

120

Z

020

4060

80100

120X

020

406080

100120

Y

X Y

Z

Fig. 2. SCN dendrite atu∞ = 0.0265, after 215 s: (a) Projection of 3D dendrite onxyplane, (b) 3D SCN dendritein the first octant.

Table 4Table shows the computational and experimental interface growth velocity for several SCN undercooling valuesin terms of (cm/s)

u∞ Microgravity vel. Rot.symmet.vel. Parallel vel.

0.04370 0.016980 0.001770 0.0018400.03380 0.008720 0.001486 0.0014800.02650 0.004620 0.001273 0.0013700.02050 0.002328 0.001066 0.0010680.01610 0.001417 0.000922 0.0009020.01260 0.000840 0.000784 0.0007560.01000 0.000500 0.000681 0.0006260.00790 0.000343 0.000590 0.000456

First two columns contain SCN undercooling values and the corresponding growth velocities from space shuttleexperiments, respectively. The computational results from parallel computing and 3D computation under therotational symmetry are given in third and fourth columns.

that have generally been used to study these problems. Furthermore the numerical schemesare also known to be reliable through various checks. Hence, it appears that the differencebetween our computations and microgravity experiments can be attributed to additionalphysics that is not part of the standard models such as (5) and (6). For the intermediatevalues of, such as 0.0126, the difference is negligible. Thus, it appears that the model (Eqs.(2) and (3)) includes the key physical components necessary to describe the solidificationprocess within this for undercooling regime. In particular, undercoolings that are at theextremes of experimental range, it is quite likely that simplest physical description givenby (5), (6), neglects physical factors that are significant in terms of the growth velocity. Atthe low end, this might include, for example, adsorption. At the high end of undercooling,


10-2000

10-3000

10-200

2

3

4

5

6

8

2

3

4

5

6

8

2

2 3 4 5

Under coding

Vel

ocity

(cm

/s)

Phase Field

IDGE

Fig. 3. Figure confirms the theoretical result that there is a linear relation between the undercooling and compu-tational growth velocity (u∞ = 0.0265).

which generates the higher velocities, the motion of the dendrite is likely to produce someconvective distribution of heat that would differ from pure heat diffusion in the liquid. Thismay be one of the source of randomness or noise that leads to extensive sidebranchingwhichwould lead to additional corrections to the velocity. At the present there are no coherentmethods to incorporate noise into the phase field (or sharp interface) equations. In theabsence of experimental data on interface noise, the use of noise in computations wouldinvolve at least two ad hoc parameters (amplitude and frequency), so that any resultingagreement with the experimental data would not be very meaningful.


Table 5Table shows the effect of kinetic coefficient on the growth velocity of interface for the undercoolingu∞ =0.0205� (s/cm2) Velocity (cm/s)

2.5× 106 0.001173.5× 106 0.000905.0× 106 0.00106

Table 6The table shows the computational interface growth velocities (cm/s) in 2D and 3D (parallel) for different under-cooling values

Undercooling 2D-velocity 3D-velocity

0.01 0.00033 0.000630.0161 0.00050 0.000900.0265 0.00066 0.00137

The model includes several features such as surface tension and kinetic undercooling.The importance of surface tension (manifested in the capillarity length,d0) has been notedin studies of linear stability[25,26]and computations. However, the kinetic undercooling,�, is often neglected in computations and theoretical studies. In fact, while the phase fieldequations naturally incorporate this material parameter, some calculations have used a limitin which � approaches zero, rather than its true value, for computational convenience. Weperform pairs of calculations for the undercooling value 0.0205 in which all parameterswere identical except for� (seeTable 5). These results indicate that a change in the kineticcoefficient of 28.5% can result in a growth velocity that is 30% as shown inTable 5. Thissuggests that� ( as := �d0 as the computation of the velocity in Gibbs–Thomson relation(7)) cannot be used as a free parameter, as can�, the interface thickness. Moreover, variationin � implies a change in growth velocity, as does a variation in any of the other parameterssuch ascv, lv, K, etc. In all other computations, we set� to be 3.5× 106 s/cm2 which isapproximated by using SCN microgravity values in (7).Our results have some interesting implications for dimensionality, as we can compare our

3D calculations to 2D calculations and to rotationally symmetric 3D calculations in whichwe used cylindrical coordinates and assumed that the dependence was purely radial. Theexperimental pictures indicate that the cylindrical symmetry of the single-needle crystalbreaks down shortly beyond the tip. Our results, however, indicate that there is relativelylittle difference in velocity between the two calculations. The tip velocity calculations in3D and 2D, on the other hand, differ by about a factor of 1.9 (seeTable 6). The ratio 1.9 canbe put in perspective by examining the limiting sharp interface Eqs. (5) and (6). Physicalintuition suggests that the growth of the interface is limited mainly by the diffusion ofthe latent heat manifested in condition (6). When diffusion is rapid, the heat equation isapproximated by Laplace’s equation, whose radial solutions are of the formrd . The latentheat condition (6) implies that the normal velocity is proportional to the gradient, ordrd−1.


0 10 20 30 40 50 60 70Seconds (s)

0.0007

0.0009

0.0011

0.0013V

eloc

ity (

cm/s

)

Fig. 4. Figure confirms the theoretical result that the growth velocity approaches a constant value in large time(u∞ = 0.0265).

Comparing this term ford =3 versusd =2, one has a ratio of 3/2=1.5.Analogously, if weexamine the Gibbs–Thomson relation alone, and solve (7) for the normal velocity, we seethat dimensionality arises (directly) in terms ofk, the sum of principal curvatures, which is(d −1)/R0 whereR0 is the radius of curvature. Hence this factor would suggest that at leastoneof the terms in this expression for the velocity hasa coefficientd−1, suggestinga ratio of(3−1)/(2−1)=2. Thus a heuristic examination of the key limiting equations suggests thatthe tip velocity in 3Dshould beabout 1.5 to 2 times that of the 2Dsystem.Of course there arenumerous nonlinearities involved in the equations that could alter this ratio.Our calculationsfall well in the range 1.5 to 2, thereby lending some support to the heuristics above. Thefully 3D calculations also allow a complete treatment of the anisotropy as it is manifestedin both directions (see Eq. (10)). While the immediate area near the tip of the dendriteappears to be symmetric about the direction of growth, the photographs of experimentsshow that there is significant asymmetry a short distance away from the tip. Consequently,there is some question as to the accuracy of rotationally symmetric computations (reducingthe 3D problem to one which is a 2D computation). However, we find that this asymmetryinfluences the tip velocity by only 8–10%.Nevertheless this anisotropy can be expected toplay a key role in the development of sidebranching for which the axial symmetry appearsto be significant.To verify the effect of the surface tension anisotropy on the interface growth, we use

four different anisotropy levels, 0.00, 0.006, 0.009 and 0.01 for the undercooling value0.0205. Corresponding growth velocities are 9.1× 10−4, 1.16× 10−3, 1.20× 10−3 and1.07×10−3 cm/s, respectively. The influence of the anisotropy on the shape of the interfaceis more clear compared to the effects on the growth velocity (see Fig.1). An order ofmagnitude change in the anisotropy strength does not change tip velocity significantlywhich confirm the experimental result[17] as well as the results from rotational symmetrycase we studied in[3]. We also observe that the tip of the interface becomes sharper in thepreferred direction as the strength of the anisotropy increases.As indicated in experiments, theory and computational studies[23], the average growth

rate of a single needle-crystal in 3D approaches a constant value. We examine this issue


by using the undercooling value,u∞ = 0.0205, for SCN. The average growth velocitiesat different time steps are calculated with the anisotropy strength�� = 0.01. The growthvelocity approaches a constant value asTfinal gets larger (seeFig. 4)which confirmspreviouscomputational and theoretical studies[23].We have performed all of our calculations on the terascale computing system, Lemieux,

at PittsburghSuperComputingCenter. Lemieux consists of 750CompaqAlphaserverES45nodes and two separate front end nodes. Each node contains four 1-GHz processors SMPwith 4 Gbytes of memory.

References

[1] T. Abel, E. Brener, H.M. Krumbhaar, Phys. Rev. E 55 (1997) 7789–7792.[2] R. Almgren, SIAM J. Appl. Math. 59 (1999) 2086.[3] Y.B. Altundas, G. Caginalp, J. Statist. Phys. 110 (2003) 1055–1066.[4] S. Balay, W.D. Gropp, L.C. McInnes, B.F. Smith, Birkhauser Press, Basel, 1997 pp. 163–202.[5] E. Ben-Jacob, N. Goldenfeld, B. Kotliar, J. Langer, Phys. Rev. Lett. 53 (1984).[6] E. Brener, V.I. Melnikov, Adv. Phys. 40 (1991) 53.[7] G. Caginalp, Carnegie Mellon Research Report, Pittsburgh, 1982.[8] G. Caginalp, Lecture Notes, Springer, Berlin, 1984.[9] G. Caginalp, Arch. Rational Mech. Anal. 92 (1986) 203.[10] G. Caginalp, Ann. Phys. 172 (1986) 136–155.[11] G. Caginalp, X. Chen, IMA Volumes on Mathematics and its Applications, vol. 43, Springer, New York,

1991, pp. 1–28.[12] G. Caginalp, E.A. Socolovsky, Appl. Math. Lett. 2 (1989) 117.[13] G. Caginalp, E.A. Socolovsky, J. Comput. Phys. 95 (1991) 85–100.[14] W.L. George, J.A. Warren, J. Comput. Physics 177 (2002) 264–283.[15] M.E. Glicksman, M.B. Koss, E.A. Winsa, Phys. Rev. Lett. 73 (1994) 573.[16] M.E. Glicksman, R.J. Schaefer, J.D. Ayers, Met. Mater. Trans. A 7 (1977) 1747–1759.[17] M.E. Glicksman, N.B. Singh, J. Crystal Growth 98 (1989) 207.[18] S. Hariharan, G.W.Young, SIAM J. Appl. Math. 62 (2001) 244.[19] G.P. Ivantsov, Dokl. Akad. Nauk USSR 58 (1947) 567.[20] A. Karma, W.-J. Rappel, Phys. Rev. E 57 (1998) 4323.[21] D.A. Kessler, J. Koplik, H. Levine, Phys. Rev. A 30 (1984) 3161;

D.A. Kessler, J. Koplik, H. Levine, Adv. Phys. 37 (1988) 255.[22] D.A. Kessler, H. Levine, Adv. Phys. 37 (1988) 255.[23] Y. Kim, N. Provatas, N. Goldenfeld, J. Dantzig, Phys. Rev. E 59 (1999) 2546.[24] M.B. Koss, M.E. Glicksman, L.T. Bushnell, J.C. LaCombe, E.A. Winsa, ISIJ 35 (1995) 604.[25] W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 34 (1963) 323–329.[26] J. Ockendon, Free boundary problems, in: E.Magens (Ed.), Proceedings of PaviaConference, Bologna, 1980.[27] Y. Pomeau, M.B. Amar, Solids far from Equilibrium, Cambridge University Press, Cambridge, England,

1991.[28] N. Provatas, N. Goldenfeld, J. Dantzig, J.C. LaCombe, A. Lupelescu, M.B. Koss, M.E. Glicksman, R.

Almgren, Phys. Rev. Lett. 82 (1999) 4496–4499.[29] L.L. Regel, W.R. Wilcox, D. Popov, F.C. Li, Acta Astronautica 48 (2001) 101–108.[30] W.A. Tiller, K.A. Jackson, J.W. Rutter, B. Chalmers, Acta Metall. 1 (1953) 428.[31] S.L.Wang, R.F. Sekerka, A.A.Wheeler, B.F. Murray, S.R. Corriell, J. Brown, G.B. McFadden, Physica D 69

(1993) 189–200.[32] J.A. Warner, R. Kobayashi, W.C. Carter, J. Crystal Growth 211 (2000).[33] A.A. Wheeler, G.B. McFadden, Proc. R. Soc. London 453A (1997) 1611.

velocity selection in 3ddendrites: phase ﬁeld computations and microgravity...

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