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A 2D stochastic micro-macro model of equiaxed eutectic solidification

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1997 Modelling Simul. Mater. Sci. Eng. 5 53

(http://iopscience.iop.org/0965-0393/5/1/004)

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Modelling Simul. Mater. Sci. Eng.5 (1997) 53–65. Printed in the UK PII: S0965-0393(97)79343-4

A 2D stochastic micro-macro model of equiaxed eutecticsolidification

Ch Charbon and R LeSarLos Alamos National Laboratory, Center for Materials Science, Mail Stop G755, Los Alamos,New Mexico 87545, USA

Received 16 August 1996, accepted for publication 29 October 1996

Abstract. We propose a model of equiaxed eutectic solidification that couples macroscopicheat diffusion with a microscopic description of nucleation and growth of the eutectic grains.The heat equation is solved numerically by means of an implicit finite difference method. Theevolution of solid fraction is deduced from a stochastic model of nucleation and growth whichuses the local temperature (interpolated from the FDM mesh) to determine the local grain densityand the local growth rate. The model predicts the evaluations of both temperature and solidfraction at any point of the sample. Moreover, a realistic appearance of the recalescence on thecooling curves, as well as a detailed picture of the microstructure, are predicted. We apply themodel to the solidification of grey cast iron.

1. Introduction

One of the major challenges in materials process modelling is the connection of microscopicmaterials properties to macroscopic processing conditions. Simulations that connect micro tomacro are rare, due to the computational difficulties inherent in coupling phenomena at suchdifferent length and time scales. Here we present a new approach for eutectic solidificationthat makes these connections and that enables detailed prediction of the microstructure froma macroscopic simulation.

Eutectic alloys are an important category of metals. Due to their low melting point,good castability and mechanical properties, they are widely used in industrial applications.Motivated by this industrial interest, scientists have long studied these alloys in order toprovide models describing the microstructural characteristics (grain density, interlamellarspacing, etc) of a casting as a function of the alloy and of the cooling conditions.

Let us consider the simple thermal situation of figure 1. A cylinder of liquid metal ofeutectic composition is insulated on its top and lateral faces and cooled at its bottom face.Solidification will proceed from the bottom of the cylinder. During solidification, threedistinct regions can be determined: a fully solid region (under line e on figure 1), a fullyliquid region (above line s in figure 1) and a mushy zone where growing solid grains arein contact with the liquid (in between line e and s in figure 1).

Equiaxed eutectic solidification occurs usually by the heterogeneous nucleation of grainsin the undercooled liquid and their subsequent growth. Due to a perfect isotropic growthvelocity, the grains grow with a spherical solid–liquid interface until they impinge on theirneighbours. From this simple description, three main phenomenon have to be accountedfor in a model of equiaxed eutectic solidification: nucleation, growth and impingement.

0965-0393/97/010053+13$19.50c© 1997 IOP Publishing Ltd 53

54 Ch Charbon and R LeSar

Figure 1. Schematics of the unidirectional solidification of a eutectic alloy. A cylindrical sampleof liquid metal is insulated on its lateral and top faces and cooled by a heat flux,Q, appliedto its bottom surface. The position of the eutectic equilibrium temperature,Teut, is indicated asa dashed line. Above line s, the metal is fully liquid. Under line e, the metal is fully solid.In between those two line, a mushy zone exists where nucleation occurs and where the grainsgrow and impinge on their neighbours.

The classical theory of heterogeneous nucleation [1, 2] fails to describe the relationshipbetween the final grain density and the maximum undercooling that is observedexperimentally [3, 4]. This is due to the presence in the melt of a population of nucleationsites with different crystallographic and chemical properties, which results in differentactivation undercoolings. The lack of a theoretical model has been circumvented by theuse of empirical relationships. Usually a set of DTA-type experiments are conducted inorder to measure the grain density obtained for different maximum undercoolings. Ananalytical relationship is then fitted to the experimental points. A number of different formsof varying complexity have been used for this analytical relationship [5–8]. A particularlysimple relation was introduced by Oldfield [3], who used the parabolic nucleation law

n(1T ) = An1T2 (1)

wheren is the grain density,1T is the local undercooling (1T = Teut− T , with Teut theequilibrium eutectic temperature andT the local temperature) andAn is a constant that hasto be determined experimentally for each system.

The growth of eutectic alloys is limited by the interlamellar diffusion of solute in theliquid ahead of the solidification front. The interlamellar spacing is in turn limited by thesurface energy associated with high curvatures. In a famous paper, Jackson and Hunt [9]derived a relationship between the growth velocity of the solidification front,v, and the

A stochastic model of eutectic solidification 55

local undercooling,

v(1T ) = Ag1T2 (2)

whereAg is a constant which depends on the thermophysical properties of the alloy. Thisrelationship, which is valid for regular eutectics, has been extended to irregular eutectics byJones and Kurz [10]:

v(1T ) =(

φ

1+ φ2

)2

4Ag1T2 (3)

whereφ is a factor characteristic of the alloy. Note that these relationships are based onthe assumption of a steady-state growth velocity. While this assumption may not be strictlycorrect, it is the best approximation available at this time and is used by most researchers[7, 8, 15–18].

Based upon the nucleation and growth relationships in equations (1)-(3), the evolutionof volumetric solid fraction,fs, is found to be

dfs

dt= n(t)4πR2

(t)v(t)9(t) (4)

wheren is the grain density,R2

is the mean square radius,v is the solid–liquid interfacevelocity and9 is the term which accounts for grain impingement. The impingement factor,9, is usually given by the model of Kolmogorov [11], Johnson and Mehl [12] and Avrami[13] as

9(t) = 1− fs(t). (5)

A review of other impingement relationships can be found elsewhere [14].Following Oldfield, a number of authors have developed deterministic models

[7, 8, 15–18] that numerically solve the heat balance of a solidifying sample, i.e.

div(κgradT ) = ρcp∂T

∂t− L∂fs

∂t(6)

whereκ is the thermal conductivity,T is the temperature,ρcp is the volumetric specificheat,t is the time,L is the volumetric latent heat andfs is the volumetric fraction of solid.

In deterministic models, the heat balance (equation (6)) is solved by a finite-differenceor finite-element method and the term∂fs/∂t on the left-hand side of the equation isdetermined explicitly from the undercoolings at the previous time step. Deterministic modelspredict the evolution of the thermal field and solid fraction as well as the grain densityand interlamellar spacing. One drawback of these models is that they produce undesiredoscillations (rebouncing of the cooling curves) in the case of the latent heat method or thatthey predict a recalescence regardless of the amplitude of the thermal gradient in the caseof the enthalpy method. Both of these numerical artifacts have been analysed by Rappaz[15] and Zou [8].

Recently, stochastic models of solidification have been developed, which track thegrowth of individual dendritic [19–23] or eutectic [14, 24, 25] grains. For nucleation andgrowth, these models are based on the same relationships as those used in deterministicmodels. The evolution of individual grains, however, is tracked by a cellular automaton[19–25] or by a mapping of the surface of the grains [14]. These models give a realisticpicture of the microstructure and therefore allow one to study microstrucural features suchas the columnar to equiaxed transition [21], grain selection [23], grain asymmetry [25] orgrain impingement [14]. They do not require any grain impingement model since it isdirectly taken into account by the competition between the grains.

56 Ch Charbon and R LeSar

In the present study, a new model for equiaxed eutectic solidification is presented, whichcouples the heat balance computation by a finite-difference method (FDM) with a stochasticmodel of nucleation and growth. The aim of the model is to predict the evolution of themicrostructure in a more accurate way than do deterministic models, to avoid rebouncing oroscillations in cooling curves and to produce a picture of the microstructure, thus allowingone to study microstructural parameters of interest.

2. The model

The principle of the model described here is that the basic coupling between the microscopicscale of the grain growth and the macroscopic scale of the heat flow is accomplished byusing in the macroscopic heat balance the solid fraction evolution predicted by a stochasticmodel of nucleation and growth. While the coupling between the microstructural level andthe macroscopic heat flow is similar to the one developed by Gandin and Rappaz [21] fordendritic solidification the microstructural models are very different. This is the first timethat a stochastic model for equiaxed eutectic solidification has been incorporated into amacroscopic calculation of solidification.

Nucleation and growth of grains are highly complex phenomena and are not wellunderstood. In this paper, we present a model for eutectic solidification that makes somesimplifying, yet reasonable approximations, which are discussed in detail below. First, weassume relatively simple relations for the nucleation rate and the growth velocity of thesolid–liquid interface. The grain growth is assumed to be spherical. We do not consider thenucleation and growth of primary austenite dendrites within the melt. Finally, we assumeno movement of the grains (i.e. fluid flow) in the course of solidification.

2.1. Macroscopic heat diffusion

We solve the heat equation in its enthalpy formulation [15] by means of a finite-differencediscretization based on a classical ‘forward in time centred in space’ implicit scheme [26].We linearize the relationship between the temperature,T , and the enthalpy,H , to get thefinal linear system [15]:[

1

1t[M] + [K]

(∂T

∂H

)t]1H = −[K]T t + bt (7)

where [K] and [M] are the conductivity and mass matrices, respectively, andbt is the vectordefining the boundary conditions,1H =H t+1t−H t . The subscriptst andt+1t indicatethat the values are taken at the present time step or at the next time step, respectively. Thediagonal matrix(∂T /∂H)t results from the linearization of theT againstH relationshipand is taken equal to 1/ρcp[I ], where [I ] is the identity matrix [21]. From the solutionof equation (7), we find the change in enthalpy during a time step at each node of themesh. Provided that the increase in solid fraction at each node,1fs, is known, the newtemperatures are given by

T t+1ti = T ti +1Hi + L1fs,i

ρcp,i(8)

where i is a subscript indicating the different nodes of the FDM mesh. The role of thestochastic method is to provide the change in solid fraction,1fs, at each node during eachtime step, from which the new temperatures can be calculated and equation (7) can besolved for the next time step. The coupling between the macroscopic heat diffusion and themicroscopic microstructural evolution takes place through equation (8).

A stochastic model of eutectic solidification 57

2.2. Microstructure evolution

The basic idea of the two-dimensional stochastic method is to use the temperature fieldpredicted by the macroscopic heat balance to drive the nucleation and growth of individualgrains at the microscopic scale. For the nucleation as well as for the growth of the grains, itis necessary to know the temperature explicitly at any point of the sample and not only at themesh nodes. The temperature at any point is simply determined by a bilinear interpolationof the temperature of the four surrounding finite-difference mesh nodes.

2.3. Nucleation

We assume an empirical nucleation law similar to that used by Oldfield [3], which givesthe grain density as a function of the undercooling,1T = T − Teut:

n(1T ) = An1Tb (9)

wheren is the grain density andAn and b are empirical parameters. We do not considernucleation and growth of primary phases (austenite or graphite) prior to the coupled eutecticgrowth.

For a given system, the local grain density at a point will be determined throughequation (9) by the maximum undercooling reached at that point, which will vary acrossthe system. Since we do not know the values of the maximum undercoolings beforehand,we choose an arbitrary value,1Tmax, that should be greater than any local undercoolingreached during a simulation. At the beginning of a calculation, we then randomly chooseNmax coordinates (xn, yn) within the sample that define possible nucleation sites. Themaximum number of nuclei in the sample,Nmax, is determined by:

Nmax= An1Tb

maxS (10)

whereS is the surface of the sample. Since the value of1Tmax is not known prior to thecomputation, a test is performed during the computation to determine whether this conditionis satisfied. If the undercooling ever exceeds the set value of1Tmax, the calculation must berestarted with a larger value or we will not predict the correct grain density. If the chosenvalue of1Tmax is too large, it will not change the predicted grain density. It will, however,increase the computational burden as we will include nucleation sites that will never beactivated.

Prior to the beginning of the calculation, an activation undercooling,1Tg, is attributedto each nuclei and is chosen randomly in the interval [0,1Tmax], with a probability forhaving a value between1T and1T + δ(1T ) given by

p(1T 6 1Tg 6 1T + δ(1T )) = dn

d(1T )

δ(1T )

Nmax= bδ(1T )1T

b−1

1T bmax

. (11)

In the time-stepping scheme, a nucleation site is activated only if its position reachesan undercooling greater than1Tg while it is still liquid (i.e. it has not been captured bya previously-nucleated grain). As soon as a nucleation site is activated, it is considered anew grain,k, with an initial infinitesimally small radius,rk.

2.4. Growth

At each time step, the grain radii are updated according to the Jackson and Hunt relationship

rt+1tk = rtk +1rtk = rtk + vtk1t = rtk + Ag(1Ttk )

21t (12)

58 Ch Charbon and R LeSar

where the local undercooling of graink, 1Tk, is determined from the undercooling atthe position of the nucleation site of the grain. In the present model, it is assumed thatthe temperature difference between the centre of a grain and any point at its solid–liquidinterface is small, so that the temperature of each grain can be considered uniform. Forthat reason, each grain is characterized by one radius (i.e. it is assumed that the solid–liquidinterface of the grains remains circular during growth).

2.5. Determination of the solid fraction increments

While the grains have circular growth, their final shape is determined by the impingementbetween growing grains. To determine this impingement, we use an approach that is verysimilar to the three-dimensional method used by Charbonet al [14] to study the influenceof grain movement on eutectic solidification. We first divide the perimeter of each grain ina large number,Ns, of equal angular sectors characterized by an angleδθ = 2π/Ns, wherethe angular position of the axis of symmetry of each sector,j , is given by the angleθj ,

θj = (j − 1)δθ j = 1, Ns. (13)

Just after the nucleation of a new grain,k, all the sectors are in contact with the liquid.As solidification proceeds, the portion of the perimeter of the grain which is in contact withthe liquid decreases due to the impingement with neighbouring grains. In order to determinethe state of each of these sectors, i.e. whether they are in contact with the liquid (and thuscan participate in further solidification) or if they have been captured by another grain, aloop over theNnear nearest neighbours is performed. To optimize the computation time, aloop is performed on theNnear neighbours of a given grain and not on all the grains presentin the sample.

As shown in figure 2, a sector,j , is considered captured if it is no longer in contact withthe liquid, i.e. if the sector of annulus defined by the two anglesθj − (δθ/2) andθj + (δθ/2)and by the two radiirk and rk + 1rk (grey area in figure 2) is intersected by one of theNnear neighbours.

If a sector is still in contact with the liquid, during growth it generates a microscopicincrease in solid fraction equal to

δf ts,k,j =δθ(rtk1r

tk + (1rtk)2/2)1S

(14)

where the numerator is equal to the surface determined by the growth of the sector (grey areaon figure 2) and where the numerator is equal to the area,1S, of the rectangle determinedby the four nearest FDM nodes surrounding the sector. This microscopic solid fractionincrement is then split in four contributions which are attributed to the four nearest FDMnodes surrounding pointS (see figure 2). The weighting factors for each contribution arethe same as those used to determine the temperature of any given point other than a meshnode (bilinear interpolation).

At each node of the finite-difference mesh, the microscopic contributions are addedand give rise, at the end of the loop over all grains and all sectors, to the macroscopicincreases of solid fraction at each node of the FDM mesh during the time step,1fs,i . It isthis macroscopic1fs,i that is used in the heat balance equation (8) to determine the newtemperatures.

A stochastic model of eutectic solidification 59

Figure 2. A grain characterized by its centre, C, and radius,r, increases in radius as a functionof time. Its impingement with its neighbours is monitored by dividing its perimeter in sectors ofangleδθ and by determining whether those sectors are still in contact with the liquid or whetherthey intersect another grain (sectors marked with an x). If a sector is still in contact with theliquid, it contributes to the solidification and generates a microscopic increment in solid fraction(grey area).

2.6. Representation of the microstructure

During the computation, the radii of the different sectors as they are captured by anothergrain are stored. Knowing the coordinates of the centre of each grain, it is then possible todraw the shape of each grain as a polygon withNs edges. The quality of the drawing isnaturally a function of the number of sectors and the duration of the time step.

3. Results and discussion

We consider here the simple geometrical and thermal situation shown in figure 1. Arectangular plate of liquid metal initially at a uniform temperature,Tini , is solidified bya constant heat flux applied on one of its small sides. The three other sides are consideredperfectly insulated. The simulation was performed with the data listed in table 1, whichwere taken from Zou [8] and correspond to grey cast iron.

The nucleation law in equation (9) was derived for three-dimensional (3D) systems.To reduce that to the two-dimensional (2D) case considered here, we use the relationshipderived by Meijering [27] to relate 2D grain densities,NA, observed on a cut through a 3Dsample, to the volumetric (3D) grain densities,Nv, i.e.

NA = 1.458N2/3v . (15)

Although this relationship was derived under the assumption of an instantaneousnucleation and constant growth rate, we believe that it is still a good approximation inthe more complicated situation considered here.

60 Ch Charbon and R LeSar

Table 1. Values of the different parameters used in the computation.

Thermal conductivity,κ 40 W m−1 K−1

Volumetric specific heat,ρcp 6.37× 106 J m−3 K−1

Volumetric latent heat,L 1.44× 109 J m−3

Eutectic temperature,Teut 1054◦CGrowth law v(1T ) = 4× 10−81T 2 m s−1

Nucleation law n(1T ) = 4.65× 1051T 1.22 m−2

Dimensions 0.03× 0.11 mMesh 13× 45 (evenly distributed) nodesInitial temperature,Tini 1350◦CHeat flux,Q 400 000 W m−2

Number of sectors,Ns 40Number of time steps,Npdt 15 000Time step,1t 0.1 s

Figure 3 shows the evolution of the microstructure in the ingot as it appears duringsolidification. The dashed line plotted across the sample shows the position of theequilibrium eutectic temperature isotherm. As time increases, the magnitude of the thermalgradient at the eutectic temperature decreases, which produces a more extended undercooledzone where liquid and solid co-exist (mushy zone). The length of the mushy zone increaseswith time, until it reaches the end of the sample.

In figure 3 we see that we have equiaxed grains throughout the sample. We define thegrain asymmetry,S, as the ratio of the distance the grain grows from the centre of nucleationalong the thermal gradient to that against the temperature gradient [25]. We measuredS

as a function of the position along the sample and found that it is roughly constant andequal to 2. If the thermal and/or nucleation conditions were changed, we could obtain moreelongated grains. In particular, if the grain density were lower, we expect that we wouldfind columnar growth near the bottom of the sample.

The model also yields the time evolution of the temperature and solid fraction. Asshown in figure 4, the shape of the cooling curves evolves from being s-shaped curves nearthe cooled surface, to classical eutectic cooling curves exhibiting the typical recalescenceand eutectic plateau near the top of the sample. Note that the apparent recalescence in thefirst curve in figure 4 is due to numerical inaccuracies due to the high gradient near thechill surface. For curves 2–10 on figure 4, note that while the curves flatten out, there isno point with dT/dt > 0, i.e. no recalescence.

The appearance of a recalescence is characteristic of low thermal gradients. We showin figure 5 the modulus of the local thermal gradient as the eutectic temperature is reached.This value varies from 10 000◦C m−1 for the locations at the bottom surface to 25◦C m−1

for the locations at the top surface. A transition occurs around 500◦C m−1. Below thisvalue of the thermal gradient, the cooling curves exhibit a recalescence whereas for thermalgradients greater than this value, they show no recalescence. It is an extraordinary featureof the stochastic model coupled with the FDM to reproduce this transition since neither theenthalpy method nor the latent heat method are able to produce such a result. The enthalpymethod coupled with a deterministic model of nucleation and growth predicts a recalescenceregardless of the amplitude of the thermal gradient and the latent heat method coupled with adeterministic model of nucleation and growth produces numerical rebouncing of the coolingcurves which can be avoided only by using an extremely short time step and/or extremelyrefined mesh [15].

A stochastic model of eutectic solidification 61

Figure 3. Simulated microstructure at different times during solidification. The dashed lineindicates the position of the equilibrium eutectic temperature. The final number of grains is37 932.

It was found that the temperature evolutions measured at different horizontal positions,x, but at the same vertical position,y, were essentially identical. Nevertheless, slightdifferences were noticed that indicate that although the macroscopic thermal situation isclearly one-dimensional, the randomness of the nucleation process induces some variationsat the microscopic scale.

We define the difference between the times at which a solid fraction of 99% and 1%were reached as the local solidification time,tf , which is plotted as a function of the positionin figure 6. We see thattf is a linear function of position at both the bottom and top of thesample, with a change in slope aty ≈ 0.08 m.

The temperature profiles at different times are plotted on figure 7. The equilibriumeutectic temperature is indicated as a dashed line on this plot. All the curves whichcorrespond to times when solidification takes place (100 s6 t 6 1200 s) exhibit a kinkwhich corresponds approximately to the beginning of the mushy zone. Att = 1000 s, thetemperature profile is located entirely under the eutectic temperature for the first time. Atthat time, the position of the kink indicating the beginning of the mushy zone is locatedaround the positiony = 0.08 m.

Thus, the discontinuity in the solidification time,tf , is at the position of the lower limitof the mushy zone when the eutectic temperature reaches the top of the sample. It can alsobe viewed as the position of the lower limit of the mushy zone when the sample ‘feels’ theadiabatic boundary.

62 Ch Charbon and R LeSar

Figure 4. Cooling curves measured at different locations,y, along the symmetry axis of thesample. The distance between two locations is equal to 1 cm with first and last curves at thebottom and top of the sample, respectively.

Figure 5. (i) Magnitude of the thermal gradient at the eutectic temperature as a function of theposition,y. (ii) Velocity of the eutectic isotherm as a function of the position,y.

The grain size,A, is plotted as a function of position in figure 8. As expected, the grainsize increases from the bottom of the sample (position of the chill) to the top (adiabaticboundary condition). The grain size is an average over a slice of 1 cm centred on theposition indicated on the horizontal axis of the plot. A transition in grain density occursaround the positiony = 0.08 m. Due to the model of instantaneous nucleation, nucleationis stopped as the recalescence is reached and no new grains appear until a temperaturebelow that of the minimum of the recalescence is reached; no nucleation occurs during thewhole eutectic plateau. Only a few grains are nucleated in the remaining liquid after theeutectic plateau is passed. In the other cases, where no recalescence is noted, nucleationcontinues regularly as long as liquid is present. The grain size as a function of the location

A stochastic model of eutectic solidification 63

Figure 6. Solidification time, defined as the time taken for the solid fraction to increase from1% to 99% , as a function of the position,y.

Figure 7. Temperature profiles at different times. The dashed line indicates the equilibriumeutectic temperature. The kinks on the temperature profiles plotted for 100 s< t < 1200 sindicate approximately the position of the beginning of the mushy zone.

is nicely fitted by a power relationship (A(y) = 6.033× 10−8 exp[6.1371(y)]). This simplerelationship is certainly a consequence of the simple thermal situation and nucleation lawused here. Only the four last points of the graph are clearly out of this fit, the transitionoccurring once more around the positiony = 0.08 m, and represent the effect of the upperboundary.

The computation time is directly related to the number of time steps, the number ofsectors and the number of nearest neighbours used for the computation, all of which haveto be relatively large to lead to a correct representation of the microstructure. Increasing thetime step and/or decreasing the number of sectors and number of nearest neighbours leadsto inaccuracies in the calculated microstructural evolution, which may lead to calculatedsolid fractions different than 1. For the present computation, setting the time step to 0.1 s,

64 Ch Charbon and R LeSar

Figure 8. Grain size as a function of the position,y. The grain size is an averageover a 1 cmwide slice. The seven first points are fitted by the exponential relationship:A(y) = 6.033× 10−8 exp[6.1371(y)].

the number of sectors to 40 and the number of nearest neighbours to 50 leads to predictedmacroscopic solid fractions in the range 1.000± 0.005. The computation burden is large,taking about 16 h on a R8000 SGI machine. The computing time could be highly decreasedby using a parallel version of the code. Since all the temperatures in the stochastic partof the model are used explicitly, it would be very easy to split the stochastic calculationamong different processors. It must also be emphasized that reasonable results can also beobtained with much less time steps and sectors. A simulation was run with only 1500 timesteps and 20 sectors in less than 3000 s leading to solid fractions in the range 1.00± 0.02.

We know of no experimental data with which we can directly compare these results.However, we must emphasize that the work described here provides a new method ofmodelling the eutectic equiaxed solidification which leads to more accurate results thanthose provided by deterministic models. In particular, the prediction of the appearance of arecalescence on the cooling curve can be compared favourably with the experimental resultspublished by Zou [8] and Rappaz [15].

4. Conclusion

A new model has been presented which couples a microscopic model of solidification withmacroscopic heat flow. This model gives a realistic picture of the microstructure during andin the end of solidification. Moreover, the model predicts cooling curves which exhibit arealistic appearance of the recalescence. Recalescences are modelled only when the thermalgradient is small and disappear when it is large.

Further experiments are necessary to go toward more quantitative comparisons. Furtherdevelopments of the model will include 3D modelling of the solidification as well as fluidflow to account for grain movement in the mushy zone due to natural or forced convection.

Acknowledgment

This work was performed under the auspices of the United States Department of Energy.

A stochastic model of eutectic solidification 65

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