A Rayleigh number based dendrite fragmentation criterion for detachment of solid crystals

during solidification

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 41 (2008) 155501 (9pp) doi:10.1088/0022-3727/41/15/155501

A Rayleigh number based dendritefragmentation criterion for detachmentof solid crystals during solidificationArvind Kumar and Pradip Dutta1

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India

E-mail: pradip@mecheng.iisc.ernet.in

Received 26 February 2008Published 26 June 2008Online at stacks.iop.org/JPhysD/41/155501

AbstractMovement of solid crystals in the form of dendrite fragments causes severe macro-segregationin solidified products. Dendrite fragmentation in the developing mushy zone occurs as a resultof remelting (causing dissolution) and subsequent breakage of dendritic side arms from thedendritic stalks. An understanding of the mechanisms of dendrite fragmentation is essentialfor predicting the transport of fragmented solid crystals for possible control ofmacro-segregation. In this work, a Rayleigh number based fragmentation criterion isdeveloped for detachment of dendrites from the developing mushy zone, which determines theconditions favourable for fragmentation of dendrites. The Rayleigh number, defined in thispaper, measures the ratio of the driving buoyancy force for the flow in the mushy zone to theretarding frictional force associated with the permeability of the mush. The criteriondeveloped is a function of the concentration difference, liquid fraction, permeability, growthrate of mushy layer and thermophysical properties of the material.

Nomenclature

C Species concentration (kg/kg)�C Concentration difference (kg/kg)D Mass diffusion coefficient of the species (m2 s−1)d1 Primary dendritic arm spacing (m)d2 Secondary dendritic arm spacing (m)G Temperature gradient (K m−1)g Acceleration due to gravity (m s−2)gl Volume fraction of liquidK Permeability (m2)kp Partition coefficientp Pressure (N m−2)r Cooling rate (K s−1)R Growth rate of the dendritic tip or Isotherm

velocity (m s−1)Ra Rayleigh numberS Source termSc Schmidt numbert Time (s)

1 Author to whom any correspondence should be addressed.

T Temperature (K)u Velocity (m s−1)�u Continuum velocity vector (m s−1)V Interdendritic fluid velocity (m s−1)

Greek symbolsβT Thermal expansion coefficient (1 K−1)βC Solutal expansion coefficientµ Dynamic viscosity (kg s−1 m−1)ν Kinematic viscosity (m2 s−1)ρ Mixture density (kg m−3)

Subscriptscr Criticali Initiall Liquid phaseo References Solid phase

1. Introduction

For alloys solidifying with a columnar dendritic microstruc-ture, a diffuse zone of porous crystalline solid, commonly

0022-3727/08/155501+09$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK

J. Phys. D: Appl. Phys. 41 (2008) 155501 A Kumar and P Dutta

known as the mushy zone, separates the melt from the fullysolidified region. The mushy zone is characterized by com-plex morphology and coupled fluid flow induced by thermaland solutal buoyancy forces, solidification shrinkage, and soon. This leads to micro- and macro-segregation, which ulti-mately affects the microstructure and mechanical propertiesof the final casting of the alloy. However, macro-segregationcan become severe if there is movement of solid crystals in theform of dendrite fragments [1–4]. The dendrite fragmentationin the mushy zone occurs as a result of remelting (causing dis-solution) and subsequent breakage of dendritic side arms fromthe dendritic stalks. Subsequently, the fluid flow in the mushyregion can transport the dendritic fragments into an under-cooled region of the melt, where they can grow to form movingequiaxed crystals, or to a region of higher temperature, wherethey can melt to form smaller particles serving as nucleationsites for equiaxed growth. These developing equiaxed crystalsmay settle or float, depending on their density relative to thatof the bulk melt, and give rise to a macro-segregation patternwhich may be different from that predicted by conventionalsolidification models without solid movement. Further, thesemoving crystals may accumulate on dendrite tips, thereby pre-cluding further advancement of a columnar dendritic front andinitiating a columnar-to-equiaxed transition (CET) [1]. Themechanism of CET, however, is not yet completely under-stood. Fragmentation of dendritic particles is considered asone of the possible mechanisms to produce CET during solid-ification [1–5]. In this respect, it is essential to have a physicalunderstanding of the remelting and fragmentation phenomenaoccurring in the mushy zone for a more accurate prediction ofthe properties of a casting.

It is well known that the presence of forced convection(e.g. electromagnetic stirring) promotes detachment [6, 7]. Inthe case of forced convection, dendrite arms bend under theapplied shear force and eventually break if a critical shear stressis reached [8]. However, detachment of dendrites can alsooccur without the presence of forced convection, if remeltingoccurs. Remelting can cause detachment of secondary ortertiary arms due to solute enrichment and thermo-solutalconvection near the dendrite root [9–11]. Several theoriesregarding remelting and detachment have been proposed inthe literature, such as the ripening process [12, 13], thermo-solutal convection in the interdendritic regions [14, 15] andlocal reheating in the mush due to release of latent heat [12,16].Experimental observations of fragmentation by side-branchremelting are reported in [13]. Recently, observations offragmentation in real alloy systems have been reported usingx-radiographic studies [9, 10, 15].

From the literature review it is evident that eithermechanical breakdown or remelting of dendrite armshas been considered to cause fragmentation. However,with natural convection alone, mechanical breakdown ofdendrites is unlikely to occur [8, 17]. In these situations,remelting of the dendrite arms during coarsening [17], soluteenrichment [5, 17, 18] and recalescence [17] have beenconsidered as possible causes of fragmentation. Although thephenomenon of dendrite fragmentation has been investigatedby several researchers, a systematic analysis for establishing

Figure 1. Schematic of the solidification problem.

a fragmentation criterion based on macroscopic parametersis yet to be found in the available literature. Establishmentof such a physics-based criterion of dendrite fragmentation isessential for identifying key parameters and predicting macro-segregation and the final microstructure of castings.

In this work, a Rayleigh number based criterion isdeveloped for detachment of solid crystals from the developingmushy zone, which determines the conditions favourable forfragmentation of dendrites. The Rayleigh number defined inthis work measures the ratio of the driving buoyancy force forthe flow in the mushy zone and the retarding frictional forceassociated with the permeability of the mush. The criteriondeveloped is a function of the net concentration difference,liquid fraction, permeability, growth rate of mushy layer andthermophysical properties. In this study, an attempt is madeto directly correlate the fragmentation phenomenon with thefactors responsible for remelting.

2. Problem description and development of dendritefragmentation criterion

We consider solidification in a rectangular cavity which isfilled with a binary alloy and cooled from the left (figure 1).At the start of the solidification, a temperature gradient, G,is imposed across the width of the cavity. The other wallsof the cavity are insulated. Solidification is initiated bylowering the temperatures of the left wall of the cavity ata constant rate, r . The configuration described above willlead to directional solidification from the left with a columnardendritic structure, once the left wall temperature falls belowthe liquidus temperature. During solidification the flow inthe melt and in the mushy zone is governed by thermally andsolutally driven convection, as shown schematically in figure 1.

The single-phase continuum momentum equation gov-erning the fluid flow in the chosen geometry (figure 1(a)) isgiven as

∂

∂t(ρ �u) + ρ �u · (∇�u)=∇ ·

(µl

ρ

ρl∇�u

)− ∇p + Sb − S �u, (1a)

whereSb = ρ �g[βT(T − T0) + βC(Cl − C0)]. (1b)

In the above formulation, the flow in the mushy zone ismodelled as in a saturated porous medium consisting of a fixedpacked bed [19]. The frictional resistance towards fluid flow

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J. Phys. D: Appl. Phys. 41 (2008) 155501 A Kumar and P Dutta

T

C

T1

T2

C1 C2

1

2

Figure 2. Schematic of the remelting process pertaining to abinary alloy.

in that region is given by the last term on the right-hand sideof equation (1a), which is modelled as [20]

S �u = µlρ �uKρl

, (2)

where K is the permeability.In this respect, two important aspects of the mathematical

description of fragmentation by remelting require specialattention: (a) the modelling of remelting phenomenon and(b) the scaling of the mushy zone velocity in terms of mushpermeability and growth rate. Assuming that the flow inthe mushy region follows Darcy’s law, the magnitude of theinterdendritic velocity, V, can be expressed as

V = �ρgK

νgl, (3)

where gl is the liquid fraction and �ρ is the net densitydifference which can be expressed as

�ρ = (ρ − ρo)

ρo= β1(Cl − Co). (4)

In equation (4), Cl is the liquid concentration and Co is areference concentration. Accordingly, the difference, �C, canbe used to determine the concentration difference. The termβ1 appearing in the above equation is the effective expansioncoefficient which can be expressed as

β1 = βC − βT · m, (5)

where m = dT/dC is the slope of the liquidus line in the phasediagram.

Local remelting in the mushy zone during solidificationcan occur in two ways: (i) by an increase in the enthalpy ofa cell due to local reheating and/or (ii) a local increase in theliquid concentration due to the flow of solute-rich liquid intothe region. Figure 2 schematically shows how remelting maytake place by these two methods. If solidification is causedby cooling from one of the mould walls, the temperature atan interior location is unlikely to increase. Since metallicalloys are generally characterized by large Lewis numbers (i.e.thermal diffusivity being much greater than solute diffusivity),

increase in the concentration at some location in the mushyzone by advection of fluid is a more likely cause for localremelting. As solidification proceeds, the solute is rejected inthe mushy region, leading to a progressive increase in the soluteconcentration. The solute concentration of the interdendriticliquid in the mushy zone will follow the liquidus line. Dueto advection of the solute-rich liquid, it is possible that thesolute concentration at a location increases isothermally, andthe point (T , Cl) in figure 2 will move to the left of the liquidusline into the liquid region of the phase diagram, suggesting thatremelting can take place. If the solidification model assumesthe lever rule, there will always be a uniform concentrationprofile in the solid, and hence no special care is required tomodel remelting. However, when using Scheil’s model (as inthe present case) which assumes no diffusion in the solid, thepresence of a frozen concentration profile in the solid requiresspecial remelting models [19, 20]. In this case, the averagesolid concentration history in each cell needs to be tracked, asdescribed in a recent study by Kumar et al [20].

Assuming the Scheil approximation, the volume fractionof liquid, gl, in the presence of the local interdendritic liquidflow, V , is given by [1]

∂gl

∂Cl= −

(1 − β

1 − kp

) (1 +

V G

r

)gl

Cl, (6)

where β is the solidification shrinkage factor (β = ρl/ρs − 1)

with ρl and ρs being densities of the liquid and solid phases,respectively, G is the temperature gradient ahead of theliquidus isotherm, r is the cooling rate at the tip and Cl isthe concentration of the liquid. The cooling rate, r , is givenas r = GR, where R is the growth rate of the mushy zone.Using the cooling rate relation given above, equation (6) canbe rewritten as

∂gl

∂Cl= −

(1 − β

1 − kp

) (1 − V

R

)gl

Cl. (7)

Now, if we neglect solidification shrinkage (i.e. if β = 0) andassume V and R to be constants, integration of equation (7)leads to

gl =(

Cl

Co

)(1/(1−kp))(1−(V/R))

. (8)

Mathematically, remelting driven by solute enrichment can becharacterized by an increase in the local liquid fraction, gl,with the increase in the local solute concentration, Cl. In otherwords, for kp < 1, remelting occurs when dgl/dCl > 0, i.e.when

V

R> 1. (9)

Equation (9) indicates that local remelting occurs whenthe interdendritic fluid velocity along the direction of thetemperature gradient becomes larger than the growth velocityof the mush (in this case, speed of the isotherms). Forarriving at a fragmentation criterion caused by remelting, twophysical situations must be considered while applying theremelting criterion given by equation (9) (refer to figure 3).Near the dendrite tips the secondary dendrite arms are notwell developed and local remelting at these locations does

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J. Phys. D: Appl. Phys. 41 (2008) 155501 A Kumar and P Dutta

x

grad T

t

x

S

L

(1) (2)

Figure 3. Schematic of dendrite fragmentation: only remelting atlocation 1; both remelting and fragmentation at location 2.

not necessarily induce fragmentation of the solid dendriticnetwork. However, deeper in the mushy zone where armsare well developed but not densely packed, local remelting atthe necks of the secondary arms can produce fragments [21].In a recent study [22], the minimum distance into the mushyzone where dendrite arms are found to be well developed isestimated to be between 6d2 and 8d2, where d2 is the finaldendrite arm spacing at the end of solidification. Based onthis, for the purpose of scaling, 8d2 is estimated to be themush depth where remelting of dendrite arms may producefragmentation. The above fragmentation criterion adopted inthis study is based only on an order-of-magnitude analysis andmay not hold accurately for all alloys under all circumstances.Accordingly, the remelting criterion given by equation (9) turnsout to be a fragmentation criterion at the mush depth, x > 8d2.

Substituting the expression for interdendritic velocityfrom equation (3) into equation (9), one obtains

�ρgK

νglR> 1. (10)

Now, choosing a length scale, l, as D/R, where D is the speciesmass diffusivity in the liquid and R is the growth rate, andsubstituting R in terms of D and l, equation (10) becomes

�ρgK

νglDl > 1. (11)

The left-hand side of equation (11) is a non-dimensionalexpression resembling a Rayleigh number, which can bedefined in the present context as

Ra = �ρgK

νglDl. (12)

Therefore, remelting occurs if

Ra = �ρgK

νglDl > 1 (13)

Table 1. Initial and boundary conditions for the base case.

Initial concentration of solute (Ci) 0.20Temperature gradient (G) 48 K mm−1

Growth rate of mush (R) 25 µm s−1

Heat extraction rate i.e. cooling 1.2 K s−1

rate at boundary (r = G · R)

and fragmentation occurs if

Ra = �ρgK

νglDl > 1 at the mush depth x > 8d2. (14)

Accordingly, we can define a critical Rayleigh number,Racr = 1, for the onset of remelting. However, this remeltingcriterion becomes a fragmentation criterion at the mush depth,x > 8d2. In other words, Racr = 1 at the mush depth, x > 8d2,for the onset of fragmentation.

The Rayleigh number, defined above, measures the ratioof the driving buoyancy force for the flow in the mushy zone tothe retarding frictional force associated with the permeabilityof the mush. It is a function of several parameters such asconcentration, liquid fraction, permeability, growth rate ofmushy layer and thermophysical properties of the alloy.

3. Results and discussion

The definition of the critical Rayleigh number for theonset of remelting given by equation (14) can be usedfor identifying the parameters influencing remelting andfragmentation phenomena. From equation (14),

Ra = f (�C, gl, g, R, K, thermo-physical properties).

(15)

The Rayleigh number definition includes the density gradientas a driving force for the flow in the mushy region, which,in turn, depends on the local concentration difference, �C.The Rayleigh number also depends on the growth rate of themush, R, which, in turn, depends on the cooling temperatureand the applied heat extraction rate (r = GR), where r

and G are the cooling rate and the temperature gradient,respectively. Thermo-solutal convection of solute in themush depends on the permeability of the mush, K . Basedon these considerations, we perform a parametric study forthe onset of fragmentation considering three major variables:namely the concentration difference (�C), the growth rate (R),and permeability (K), which depends on the liquid fraction(gl). For this case study, we consider solidification of anAl–20 wt% Cu alloy. The data for thermophysical propertiesare taken from [7]. A base case is chosen for which theinitial and boundary conditions are given in table 1. Thealloy composition and the range of cooling rate for the basecase are chosen such that the conditions are favourable forfragmentation, as observed in the experiments reported in [10].

To appreciate this methodology, we have plotted Ra asa function of each parameter, by keeping other parametersconstant. Each plot gives the range of parameters withinwhich fragmentation is likely to occur, based on the criticalRa definition.

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J. Phys. D: Appl. Phys. 41 (2008) 155501 A Kumar and P Dutta

Figure 4. Variation of Ra with the concentration difference (�C): (a) for different liquid fractions and with R = 25 × 10−6 m s−1,G = 48 K mm−1, r = 1.2 K s−1, (b) for different R and with gl = 0.8, G = 48 K mm−1, (c) for different Sc and with gl = 0.8,R = 25 × 10−6 m s−1, G = 48 K mm−1, r = 1.2 K s−1 and (d) for different gravities and gl = 0.8, R = 25 × 10−6 m s−1, G = 48 K mm−1,r = 1.2 K s−1.

3.1. Effect of concentration difference (�C)

Figure 4(a) shows the variation of the Rayleigh number withthe concentration difference (�C) for two different liquidfraction values. The remelting and fragmentation regimes arealso marked in the figure. The depth of 8d2 in the mushy zone,where remelting can cause fragmentation, corresponds to theconcentration difference (�C) of 0.04. It is observed that fora lower liquid fraction value, a higher concentration differenceis required for remelting to take place. In other words, for agiven concentration difference, a more permeable mush is morelikely to remelt and fragment. Figure 4(b) shows the variationof the Rayleigh number with the concentration difference (�C)for two different values of growth rates, R. This plot showsthat higher growth rates will require a greater concentrationdifference (�C) for the onset of remelting. In other words, fora given concentration difference, higher solidification growthrates lead to a less likelihood of remelting and fragmentation.The effect of the Schmidt number, Sc, representing differentalloy systems, is shown in figure 4(c). For higher Sc numberalloys, remelting is more likely to initiate, and hence thosesystems are more prone to fragmentation even with a lessconcentration difference. The effect of gravity is shown in

figure 4(d). It can be seen that the reduced gravity conditionscan prevent remelting and fragmentation.

3.2. Effect of growth rate

Figure 5(a) shows the variation of the Rayleigh numberwith the growth rate for different values of the concentrationdifference (�C). The same conclusion, as inferred fromfigure 4(b), can be drawn that higher growth rates makeremelting difficult. However, the concentration difference actsas a positive feedback for remelting. Figure 5(b) shows that forthe liquid fraction equal to 0.8, an increase in the growth rateresults in a decreasing tendency for remelting. However, forthe liquid fraction equal to 0.9, the mush undergoes remeltingfor all growth rates. In other words, a more permeable mushis more likely to undergo remelting and hence fragmentation.

3.3. Effect of permeability

The permeability term appearing in equation (3) is accordingto the Carman–Kozeny relationship and is given as [20]

K = d22 g3

l

180(1 − gl)2. (16)

5

J. Phys. D: Appl. Phys. 41 (2008) 155501 A Kumar and P Dutta

Figure 5. Variation of Ra with the growth rate R (a) for differentconcentration differences (�C) and with gl = 0.8, G = 48 K mm−1

and (b) for different gl and with concentration difference,�C = 0.04, G = 48 K mm−1.

The above relationship loses accuracy close to the dendrite tips,where the permeability is a function of the primary as well assecondary arm spacing [23]. Nevertheless, it can still give areasonable estimate of flow penetration in the mushy zone forboth alloys. Similarly, it is understood that d2 can vary fromthe dendrite tips to the roots, but a fixed value correspondingto the secondary arm spacing measured experimentally at theend of solidification for each alloy has been used. In this work,the value of d2 is taken as 1 µm.

As permeability is a function of the liquid fraction, thevariation of the Rayleigh number with the liquid fraction isplotted (figure 6). It is assumed that the vertical line at theliquid fraction equal to 0.8 corresponds to a mush depth of 8d2,beyond which remelting leads to fragmentation. Figure 6(a)shows that the mush should be sufficiently permeable forremelting to take place (gl > 0.75). There is a very smallregime in which fragmentation can take place for the higherconcentration difference (�C) value of 0.1. However, alower value of �C = 0.02 does not cause fragmentation,although remelting is possible. In figure 6(b), it is observed

Figure 6. Variation of Ra with liquid fraction (a) for differentconcentration difference (�C) and with R = 25 × 10−6 m s−1,G = 48 K mm−1, r = 1.2 K s−1 and (b) for different R and withconcentration difference, �C = 0.04, G = 48 K mm−1.

that lower growth rates are required for both remeltingand fragmentation to take place. There is a very smallregime in which fragmentation can take place for the lowergrowth rate of 10 µm s−1. However, a higher growth rate of25 µm s−1 does not cause fragmentation, although remeltingis possible.

Figure 7 shows a map demarking the remelting andfragmentation zone for various values of interdendriticvelocities and mush permeability in terms of the liquid fraction.It is evident from this figure that for interdendritic velocitiesless than 25 µm s−1, which correspond to Racr < 1, there isno possibility of remelting at any location in the mush. Thevarious values of liquid fractions in the above plot correspondto different mush depths. It is assumed that the vertical lineat a liquid fraction of 0.8 corresponds to a mush depth of8d2, beyond which remelting leads to fragmentation. Thisplot gives an interesting result in the form of a relationshipbetween interdendritic velocity and liquid fraction describingthe remelting and fragmentation zone in the mush.

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J. Phys. D: Appl. Phys. 41 (2008) 155501 A Kumar and P Dutta

Figure 7. Variation of the interdendritic velocity with liquidfraction. The fragmentation and remelting maps are shown forvarious interdendritic velocities in the mush. Concentrationdifference, �C = 0.04, R = 25 × 10−6 m s−1, G = 48 K mm−1,r = 1.2 K s−1.

Anisotropic permeability. Permeability is a complex param-eter, which depends on the local liquid fraction (gl) and thedendritic arm spacings [23]. In the past, several authorshave conveniently used an isotropic permeability given by theKozeny–Carman approximation [24]. It is recognized that theisotropic permeability given by the Kozeny–Carman approxi-mation [24] is valid for the solid fraction up to about 0.5. Foran anisotropic variation, the permeability used in the principaldirections can be calculated using the primary dendritic armspacing, as reported by Heinrich and Poirier [19]:

Kx =

1.09×10−3g3.32l d2

1 gl �0.65,

4.04×10−6

[gl

1−gl

]6.7336

d21 0.65<gl �0.75,(

−6.49×10−2 +5.43×10−2

[gl

1−gl

]0.25

d21

)

0.75<gl �1.0,

(17)

Ky =

3.75×10−4g2l d

21 gl �0.65,

2.05×10−7

[gl

1−gl

]10.739

d21 0.65<gl �0.75,

0.074×[log(1−gl)−1.49+2(1−gl)−0.5(1−gl)2]d2

1

0.75<gl �1.0.

(18)

The components of the anisotropic permeability along withisotropic variation are shown in figure 8 for the primarydendritic arm spacing, d1 = 100 µm. Figure 8 showsthat the Carman–Kozney isotropic permeability variationoverestimates the permeability for most liquid fractions, incomparison with the anisotropic permeability variation. Theprimary dendritic arm spacing, d1, appearing in the anisotropicvariation of permeability, is not a constant. Instead, it depends

Figure 8. Permeability variation with the volume fraction of liquidfor d1 = 100 µm and d2 = 1 µm.

Figure 9. Variation of primary dendritic arm spacing, d1, withliquid concentration.

on several parameters as given by Tewari and Shah [25]:

d1 = 921.7 × C0.134G−0.354R−0.261. (19)

This variation of the dendritic arm spacing d1 (µm) withthe liquid concentration for a given temperature gradientG (K cm−1) and growth rate R (µm s−1) has a significantinfluence on the Rayleigh number. Figure 9 shows the variationof d1 with the liquid concentration, C. The variation of theRayleigh number as a function of the liquid concentration andthe liquid fraction is shown in figure 10. Figure 10(a) showsthe Rayleigh number variation with constant d1, while thecase with a variable d1 is plotted in figure 10(b), in whichequation (19) is used to account for dendritic arm spacingvariation. It is observed that the maximum value of theRayleigh number for the constant d1 case is nearly 50% morethan that with a variable d1. Hence, the dependence of dendriticarm spacing on the liquid concentration plays an important

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J. Phys. D: Appl. Phys. 41 (2008) 155501 A Kumar and P Dutta

Figure 10. Variation of Ra with gl and Cl withR = 25 × 10−6 m s−1, G = 48 K mm−1, r = 1.2 K s−1

(a) constant d1 (b) d1 = f (Cl)

role in the onset of remelting and further fragmentation in themushy zone.

4. Conclusions

A Rayleigh number based dendrite fragmentation criterionis proposed for a better understanding and prediction of thecomplex interactions during remelting and fragmentation inthe developing mushy zone. The fragmentation criteriondeveloped in this paper is a function of the concentrationdifference, liquid fraction, permeability, growth rate of mushylayer and thermophysical properties. The idea is stronglysupported by the fact that, when this criterion is quantified interms of all dependent parameters, it can (i) explain the onset ofremelting and fragmentation in the developing mush pertainingto various cases, (ii) predict a range of various parametersleading to remelting and fragmentation and (iii) estimatethe remelting and fragmentation regimes. Specifically, it isfound that a more permeable mush is more likely to remeltand fragment. Higher solidification growth rates lead to theless likelihood of remelting and fragmentation in the mush.For higher Sc number alloys, remelting is more likely toinitiate and hence they are more prone to fragmentation. Lowgravity conditions will reduce the chances of remelting andfragmentation. It is also shown that for the present case, alower concentration difference does not cause fragmentation,

although remelting is observed. Finally, it is observed thatthe anisotropic variation of permeability and the dependenceof dendritic arm spacing on the liquid concentration playan important role in the onset of remelting and subsequentfragmentation in the mushy zone.

Acknowledgment

The authors gratefully acknowledge the financial support ofGeneral Motors Corporation, USA.

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