Available online at www.sciencedirect.com

www.elsevier.com/locate/actamat

Acta Materialia 57 (2009) 607–615

Predicting the columnar-to-equiaxed transition for a distributionof nucleation undercoolings

M.A. Martorano *, V.B. Biscuola

Department of Metallurgical and Materials Engineering, University of Sao Paulo, Av. Prof. Mello Moraes, 2463 Sao Paulo-SP 05508-900, Brazil

Received 19 March 2008; received in revised form 29 September 2008; accepted 1 October 2008Available online 29 October 2008

Abstract

A deterministic mathematical model for steady-state unidirectional solidification is proposed to predict the columnar-to-equiaxedtransition. In the model, which is an extension to the classic model proposed by Hunt [Hunt JD. Mater Sci Eng 1984;65:75], equiaxedgrains nucleate according to either a normal or a log-normal distribution of nucleation undercoolings. Growth maps are constructed,indicating either columnar or equiaxed solidification as a function of the velocity of isotherms and temperature gradient. The fieldsof columnar and equiaxed growth change significantly with the spread of the nucleation undercooling distribution. Increasing the spreadfavors columnar solidification if the dimensionless velocity of the isotherms is larger than 1. For a velocity less than 1, however, equiaxedsolidification is initially favored, but columnar solidification is enhanced for a larger increase in the spread. This behavior was confirmedby a stochastic model, which showed that an increase in the distribution spread could change the grain structure from completely colum-nar to 50% columnar grains.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Columnar-to-equiaxed transition; Directional solidification; Casting; Aluminum alloys

1. Introduction

The columnar-to-equiaxed transition (CET) is the tran-sition from columnar grains to equiaxed grains observed inthe macrostructures of castings. The position of the CETdetermines the relative amount of columnar and equiaxedgrains, which defines important properties of cast products[1,2]. Different mechanisms leading to the CET have beenproposed, investigated and reviewed in the past decades[3–5]. There is a consensus that the CET occurs when themoving front of columnar grains is blocked by equiaxedgrains growing in the undercooled liquid ahead of thisfront. This mechanism has been observed in situ using syn-chrotron X-ray radiographs [4]. There is still controversy,however, about the details of how the equiaxed grains stopthe columnar front. In the first mechanism proposed, equi-axed grains block the columnar front through mechanical

1359-6454/$34.00 � 2008 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2008.10.001

* Corresponding author. Tel.: +55 11 3091 6032; fax: +55 11 3091 5243.E-mail address: martoran@usp.br (M.A. Martorano).

interactions (mechanical blocking) [3]. Later, equiaxedgrains were assumed to block columnar grains through sol-utal (solutal blocking) [6] or thermal (thermal blocking)interactions [7,8]. The mechanical and solutal blockingmechanisms were considered simultaneously in a recentmathematical model to predict the CET [9].

Regardless of the blocking mechanism, experiments andtheoretical models have shown that the CET is significantlyaffected by the number density of equiaxed grains and thenucleation undercooling. In Hunt’s [3] model for unidirec-tional steady-state solidification, a decrease in the nucle-ation undercooling or an increase in the number ofequiaxed grains facilitates the blocking of columnar grainsand, consequently, the occurrence of the CET. Sturz et al.[10] and Martorano and Capocchi [11] added inoculants tomelts of Al and Cu alloys, respectively, observing that theCET occurred earlier in comparison with samples withoutinoculants, confirming the predictions of Hunt’s [3] model.

To predict the CET, instantaneous nucleation wasadopted in earlier models [3,6] by assuming that all possible

rights reserved.

608 M.A. Martorano, V.B. Biscuola / Acta Materialia 57 (2009) 607–615

equiaxed grains would nucleate locally when the liquidundercooling was larger than DTN = TL � TN, where TL

is the liquidus temperature of the alloy and TN is a givennucleation temperature. Nevertheless, in models concernedwith the prediction of equiaxed grain size, the nucleationundercooling was assumed to follow a Gaussian distribu-tion. This distribution can be interpreted as a distributionof undercoolings necessary for heterogeneous nucleationon different substrate particles. The distribution density,dn/dDTN, was defined as [12]:

dndðDT N Þ

¼ nTffiffiffiffiffiffi2pp

� DT r

exp � 1

2

DT N � DT N

DT r

� �2" #

; ð1Þ

where n is the number density of substrate particles; nT isthe number density of all substrate particles in the system;DT N is the average nucleation undercooling; and DTr is thestandard deviation of the distribution. For instantaneousnucleation, DTr = 0, and the distribution density shouldbe replaced with dn=dDT N ¼ nT dðDT N Þ, where d is the Dir-ac delta function.

Experimental evidence has shown that the undercoolingnecessary for a heterogeneously nucleated grain to growfreely from its substrate particle depends on the particlesize, /, according to [13]:

DT fg ¼4rSL

DSV /; ð2Þ

where DTfg is the undercooling for free growth; rSL is thesolid–liquid interface energy per unit area; DSV is the vol-umetric entropy of fusion. In equiaxed grain solidification,this free growth undercooling can be modeled as a nucle-ation undercooling [13]. This undercooling distribution de-pends on the particle size distribution, which was assumedto be either exponential [13] or log-normal [14]:

dnd/¼ nT

r//ffiffiffiffiffiffi2pp exp � 1

2

lnð/=�/Þr/

� �2" #

; ð3Þ

where �/ and r/ are, respectively, the geometric average andthe shape parameter of the log-normal distribution.

In most of the deterministic models of the CET, instan-taneous nucleation has been assumed [3,6,15–18]. Wu andLudwig [9], however, have recently considered a Gaussiandistribution of undercoolings in their transient model, butthe effects of the distribution spread on the CET positionwere not investigated. Quested and Greer [14] developeda steady-state model considering a log-normal distributionof nucleant particle sizes to predict equiaxed grain size. Themodel was extended to predict the CET, but the effects ofthe distribution spread on the CET were not examined.Nevertheless, the effects of the distribution spread on grainsize were investigated to design inoculants for aluminumalloys [19,20].

A distribution of nucleation undercoolings has alwaysbeen adopted in stochastic models [21–23], but only Choet al. [23] studied the effects of the spread of a Gaussian

distribution on the calculated grain macrostructures. Theeffects on the CET position, however, were not discussed.

Except for Greer and Quested’s [14] model, all determin-istic and stochastic models that predict the CET and con-sider a nucleation undercooling distribution have beentransient. Although transient models are useful for predict-ing the grain structure in solidification processes, thesteady-state models have proved very important to theunderstanding of the basic mechanisms of the CET [3],since the velocity of isotherms, V, and the temperaturegradient, G, are changed independently. Furthermore,steady-state models can be readily used to construct growthmaps, and have been frequently applied to predict eitherthe prevailing growth mode (columnar or equiaxed) insteady-state solidification [24] or the CET in transientsolidification [25]. Transient models can also be used toconstruct the maps, but numerous time-consuming simula-tions (which need to be carried out for long enough toachieve steady state) are frequently required [6,9,22].

In the present work, a steady-state model of unidirec-tional solidification is proposed to predict the CET. In thismodel, equiaxed grains nucleate within a temperaturerange, differing from several available models consideringinstantaneous nucleation. The model presented here isbased on the model presented by Quested and Greer [14]and is an extension of Hunt’s [3] model (in which instanta-neous nucleation was assumed) to a more realistic situationof equiaxed grains nucleating within a temperature range.

Mathematical models have shown that the CETpositions predicted with either the mechanical or solutalblocking mechanism are similar for most solidification con-ditions [6,26]. Therefore, mechanical blocking of thecolumnar front is adopted in the present work, becausethe construction of the growth maps is facilitated afterthe derivation of a simple final equation. Growth mapsare constructed to predict the CET assuming that the distri-bution of nucleation undercoolings for equiaxed grains iseither Gaussian or log-normal. The effects of the spreadof the distributions are analyzed and verified in the macro-structures calculated with a well-known stochastic model ofsolidification.

2. Description of the model

The present model, which is based on the models pro-posed by Hunt [3] and Quested and Greer [14], is developedfrom the following assumptions: (a) unidirectional andsteady-state solidification of both equiaxed and columnargrains for a reference system moving at the constant veloc-ity of the isotherms; (b) linear temperature variation withdistance; (c) negligible convection of the liquid and move-ment of equiaxed grains; (d) spherical equiaxed grains; (e)heterogeneous nucleation of equiaxed grains on substrateparticles; and (f) normal (Gaussian) or log-normal distribu-tion of nucleation undercoolings. The solidification systemis illustrated in Fig. 1, where both columnar and equiaxedgrowth are depicted, although only one type of growth

TLT

iNT

gR

gV

gε

V

iNTΔ1i

NT +ΔcolTΔ 1i

NT +

a

b

V

isotherms

columnar equiaxed

( )colg TΔε

Fig. 1. Unidirectional steady-state solidification system: (a) columnarfront grows with the velocity of isotherms, V; equiaxed grains nucleate atdifferent undercoolings and grow with radial velocity Vg to radius Rg; (b)profiles of volume fraction of equiaxed grains (eg) and temperature,indicating the liquidus temperature (TL), nucleation temperatures (TN)and undercoolings (DTN) on different substrate particles i and i + 1, andthe columnar front undercooling (DTcol).

M.A. Martorano, V.B. Biscuola / Acta Materialia 57 (2009) 607–615 609

prevails under steady-state conditions. The two types ofgrowth are modeled separately as described below.

In equiaxed solidification, consider an equiaxed grainnucleating at time tN, at a location of undercooling DTN

observed from a reference fixed out of the system, i.e. notmoving with the isotherms. At time t > tN, the undercool-ing at this location is DT > DTN and the radius of the equi-axed grain, Rg, is [3]:

RgðDT N ;DT Þ ¼Z t

tN

V gdt ¼Z DT

DT N

ADT m

VGdDT

¼ AVGðmþ 1Þ ðDT mþ1 � DT mþ1

N Þ; ð4Þ

where Vg is the radial growth velocity of the equiaxedgrain; V is the velocity of the isotherms, which equals thecolumnar front velocity when columnar growth prevails;and G is the temperature gradient. To derive the right-handside of Eq. (4), the integration variable was first changedand the hypotheses of steady-state and linear temperatureprofile were used to calculate the cooling rate as the prod-uct V G. The radial growth velocity was:

V g ¼ ADT m; ð5Þwhere constants A and m depend on the alloy and can beobtained either from experiments or from curve fitting toa more elaborate dendritic growth model.

Provided the impingement of equiaxed grains isneglected, at the local undercooling DT the volume fraction

of the equiaxed grains that nucleated between DTN andDTN + dDTN is:

degE ¼4

3pR3

gðDT N ;DT Þdn ¼ 4

3pR3

gðDT N ;DT Þ dndDT N

dDT N ;

ð6Þwhere egE is the extended volume fraction of equiaxedgrains. If all equiaxed grains that nucleated in the underco-oling range 0 6 DTN 6 DT are accounted for, then:

egEðDT Þ ¼Z DT

0

4

3pR3

gðDT N ;DT Þ dndDT N

dDT N : ð7Þ

Finally, correcting for grain impingement (Avrami cor-rection) yields the total volume fraction of equiaxed grainsat a given undercooling, DT, as follows:

egðDT Þ ¼ 1� expð�egEðDT ÞÞ: ð8ÞConsider now only columnar solidification. At steady

state, the columnar front velocity equals that of the iso-therms, V, and the undercooling (in relation to the liquidustemperature) at the front position is DT = DTcol. Next, thestrategy adopted by Hunt [3] to determine the fields ofcolumnar or equiaxed solidification in the growth maps isadopted in the present work: columnar and equiaxedgrowth are superimposed on the linear temperature profile(Fig. 1) and, if the volume fraction of equiaxed grains atthe columnar front undercooling (eg(DTcol)) is larger thana predetermined blocking fraction (eblock

g ), equiaxed growthprevails. Otherwise, columnar grains dominate. Thisassumption implies the mechanical blocking of the colum-nar front. Mathematical models have predicted that bothmechanical and solutal blocking of the columnar frontyield similar results in most solidification conditions[6,26]. Consequently, the mechanical blocking criterion isadopted, because the final model equation is simplifiedand growth maps are readily calculated.

The boundary curve between columnar and equiaxedsolidification in the growth maps can finally be calculatedby considering egðDT colÞ ¼ eblock

g together with Eqs. (4),(7) and (8), giving:

G ¼ 0:617n1=3T uðDT col; e

blockg ÞDT col ð9Þ

uðDT col; eblockg Þ ¼ 3

ðmþ 1Þ0:66

lnð1� eblockg Þ�1

Z DT Col

0

(

1� DT N

DT col

� �mþ1" #3

1

nT

dndDT N

dDT N

9=;

1=3

:

ð10Þ

When G is lower than the value given by Eq. (9), graingrowth is equiaxed; otherwise, it is columnar. Eqs. (9)and (10), which can be integrated numerically by simplemethods, can be used to calculate the growth maps for agiven probability density function, dn/dDTN. For instanta-neous nucleation, dn=dDT N ¼ nT dðDT NÞ, where DT N is the

610 M.A. Martorano, V.B. Biscuola / Acta Materialia 57 (2009) 607–615

average nucleation undercooling. If also m = 2 and

eblockg ¼ 0:49, then u ¼ 1� DT N

DT col

� �3

, which gives Hunt’s[3]

model when substituted in Eq. (9).Provided a blocking fraction is defined, Eqs. (9) and (10)

allow calculation of the boundary curve between columnarand equiaxed growth in growth maps with DTcol and G asthe independent variables. To substitute DTcol for V, creatinga more useful map, the columnar front is assumed to growaccording to the same law as that for equiaxed grains (Eq.(5)), resulting in DTcol = (V/A)�m. Using this relation, Eqs.(9) and (10) can be, respectively, written in the followingdimensionless form:

G� ¼ 0:617u�ðV �; eblockg ÞV �1=m ð11Þ

u�ðV �; eblockg Þ ¼ 3

ðmþ 1Þ0:66

lnð1� eblockg Þ�1

Z V �1=m

0

(

1� DT �NV �1=m

� �mþ1" #3

dn�

dDT �NdDT �N

9=;

1=3

; ð12Þ

where G� ¼ G=ðn1=3T DT N Þ; V � ¼ V =ðADT N

mÞ; DT �N ¼ DT N=DT N ; and n* = n/nT. The parameter DT N , a scale for any typeof undercooling in the model, can be chosen as the average ofthe distribution of nucleation undercoolings. Note that theapplication of Eq. (5) to calculate the columnar front veloc-ity (which equals the isotherm velocity at steady state) givesDT �col ¼ ðV �Þ

1=m, corresponding to the integration limit inEq. (12). To recover Hunt’s model [3] (instantaneous nucle-ation) u* = 1 � (V*)�3/m.

Either a normal or a log-normal distribution of nucle-ation undercoolings was adopted to integrate Eq. (12).The dimensionless form of the normal distribution was

dn�

dDT �N¼ 1

DT �rffiffiffiffiffiffi2pp exp � 1

2

DT �N � 1

DT �r

� �2" #

; ð13Þ

where DT �r ¼ DT r=DT N . Then, Eq. (12) becomes:

u� ¼ 3

ðmþ1Þ0:66

lnð1� eblockg Þ�1

1

DT �rffiffiffiffiffiffi2pp

Z V �1=m

0

(

� 1� DT �NV �1=m

� �mþ1" #3

exp �1

2

DT �N �1

DT �r

� �2" #

dDT �N

9=;

1=3

:

ð14ÞThe extended volume fraction of equiaxed grains at any

undercooling DT* was also calculated for a normal distri-bution of nucleation undercoolings combining Eqs. (7)and (13), yielding:

egEðDT �Þ ¼ 2

3

ffiffiffiffiffiffi2pp

DT �r

1

½V �G�ðmþ 1Þ�3Z DT�

0

ðDT �Þmþ1 � ðDT �N Þmþ1

h i3

exp � 1

2

DT �N � 1

DT �r

� �2" #

dDT �N : ð15Þ

Finally, Eq. (8) was used to correct egE, giving eg(DT*).As discussed previously, the distribution of undercoo-

lings for free growth depends on the log-normal distribu-tion of particle sizes (Eq. (3)). Using Eq. (2), the sizedistribution given by Eq. (3) was converted into the follow-ing dimensionless undercooling distribution:

dn�

dDT �N¼ 1

r/DT �Nffiffiffiffiffiffi2pp exp � 1

2

lnðDT �N Þr/

� �2" #

; ð16Þ

where the undercooling scale was the geometric averageDT N ¼ 4rSL

DSV /, which is a function of the geometric average

of the size distribution (�/Þ from Eq. (3). Note that theshape parameter (r/), which is different from the standarddeviation, is already dimensionless. It indicates the shape,rather than the spread of the distribution, but is relatedto the standard deviation. Substituting Eq. (16) into Eq.(12) gives:

u� ¼ 3

ðmþ1Þ0:66

lnð1� eblockg Þ�1

1

rffiffiffiffiffiffi2pp

Z V �1=m

0

(

� 1� DT �NV �1=m

� �mþ1" #3

1

DT �Nexp �1

2

lnðDT �N Þr

� �2" #

dDT �N

9=;

1=3

;

ð17Þ

which can be used to calculate the growth maps for a log-normal distribution of undercoolings when combined withEq. (11).

3. Growth maps for steady-state solidification

The boundary curves between columnar and equiaxedsolidification in the growth maps were constructed by com-bining Eq. (11) with either Eq. (14), for the normal distri-bution of undercoolings, or Eq. (17), for the log-normaldistribution. The maps were constructed for an Al–7 wt.% Si alloy with the normal (Fig. 2a) and log-normalundercooling distributions (Fig. 2b). The Lipton–Glicks-man–Kurz (LGK) curve presented by Martorano et al. inFig. 12 of Ref. [6] is also included. This curve is shownbecause it was calculated with a model similar to that pro-posed by Hunt [3], in which instantaneous nucleation wasassumed, but considering the LGK model [27] for dendriticgrowth. The LGK model has been adopted in numerousmathematical models of the as-cast grain macrostructure,representing a reference model for dendritic growth. Allcurves are plotted in the more familiar dimensional form.

The boundary curves for both distributions reduce tothe instantaneous nucleation curve (LGK) when DTr andr/ decrease to 0.01 K and 0.01, respectively. This behaviorwas expected, since the normal and log-normal distribu-tions of undercoolings approach the instantaneous nucle-ation model when DTr ? 0 and r/ ? 0, respectively.

When V b10�4 m s�1, an increase in DTr and r/ fromthe instantaneous nucleation values shifts the boundarycurves to the right-hand side, widening the field of equiaxedsolidification. For a larger increase (DTr > 1 K, r/ > 3 K),

Fig. 2. Growth maps of isotherm velocity (V) vs. temperature gradient (G)for: (a) normal distribution of nucleation undercoolings with differentstandard deviations, DTr; and (b) log-normal distribution of undercoo-lings with different shape parameters, r/. The LGK curve was obtained byMartorano et al. [6] for instantaneous nucleation conditions. In thecalculations DT N ¼ 3 K, m = 2.7 and eblock

g ¼ 0:49.

Fig. 3. Growth maps of dimensionless isotherm velocity (V*) andcolumnar front undercooling (DT �col) vs. dimensionless temperaturegradient (G*) for: (a) normal distribution of nucleation undercoolingswith different standard deviations, DT �r; and (b) log-normal distribution ofnucleation undercoolings with different shape parameters, r/. Twosolidification conditions (A and B) are also indicated by dots. In thecalculations m = 2.7 and eblock

g ¼ 0:2.

M.A. Martorano, V.B. Biscuola / Acta Materialia 57 (2009) 607–615 611

the curves begin to shift to the left, enhancing the columnarsolidification fields again. For V a 10�4 m s�1, however,the curves always shift to the left-hand side, favoringcolumnar growth. A similar behavior is observed in thedimensionless growth maps of Fig. 3, where a DT �col axis(related to the V* axis) is also included. In these mapseblock

g ¼ 0:2 was adopted, rather than the traditional 0.49proposed by Hunt [3], because it has recently been shownto improve the agreement between the CET positionsobtained with stochastic and deterministic models [26].

Solidification conditions for V* > 1 (condition A:V* = 10, G* = 1.6) and V* < 1 (condition B: V* = 0.1, G* =

0.02) are indicated by dots in Fig. 3a and examined inFig. 4, where eg is given as a function of DT* for a normaldistribution of nucleation undercoolings. As expected, eg

increases with an increase in DT* during solidification.For condition A (Fig. 4a) and approximately instantaneousnucleation (DT �r ¼ 0:01), equiaxed grains nucleate in a nar-row undercooling range around DT* = 1 (centre of thedimensionless normal distribution in Fig. 5a). The grainfraction reaches the blocking fraction (eblock

g ¼ 0:2) exactlyat the columnar front undercooling (DT* � 2.3), as shownin Fig. 4a. Consequently, condition A is at the boundarycurve for DT �r ¼ 0:01 (Fig. 3a) and also for DT �r ¼ 0:2.

Fig. 4. Volume fraction of equiaxed grains (eg) as a function ofdimensionless undercooling (DT*) for several standard deviations (DT �r)and two conditions of temperature gradient (G*) and isotherm velocity(V*): (a) G* = 1.6, V* = 10 (condition A); (b) G* = 0.02, V* = 0.1(condition B). The blocking fraction (eblock

g ¼ 0:2) and the columnar frontundercooling (DT �col) are indicated.

Fig. 5. Probability densities, dn�=dDT �N , for the distributions of nucleationundercoolings, DT �N , in dimensionless form: (a) normal distributions fordifferent standard deviations, DT �r; and (b) log-normal distributions fordifferent shape parameters, r/.

612 M.A. Martorano, V.B. Biscuola / Acta Materialia 57 (2009) 607–615

When DT �r increases from 0.2 to a value in the rangebetween 1 and 10, then eg < eblock

g at DT �col (Fig. 4a), andcolumnar growth prevails, as observed in the growth map.The spread of the distribution of undercoolings (Fig. 5a)increases, reducing the maximum number of equiaxed grainsthat can nucleate in the range 0 < DT � < DT �colð� 2:3Þ. Thismaximum number is proportional to the integrated areaunder the probability density curve from the liquidus iso-therm (DT* = 0) to the columnar front isotherm(DT* � 2.3). When DT �r is between 1 and 10, a significant por-tion of the area is cut off on the left-hand side by the liquidusisotherm and on the right-hand side by the columnar frontisotherm. Consequently, the equiaxed grain fraction at

DT �col decreases owing to fewer equiaxed grains, favoringcolumnar growth. Generally, the area under the normal dis-tribution curve is significantly reduced by the liquidus iso-therm when DT �r > 1=3, and by the columnar frontisotherm when DT �r > ðDT �col � 1Þ=3, which is 0.43 for condi-tion A (DT �col ¼ 2:3).

Solidification condition B represents a point on theboundary curve for DT �r ¼ 0:2 (Fig. 3a), because eg ¼ eblock

g

at the columnar front undercooling (DT* � 0.43), as shownin Fig. 4b. When DT �r decreases to 0.01 (approximatelyinstantaneous nucleation) all equiaxed grains nucleate atabout DT* = 1. Thus, no equiaxed grains nucleate belowDT �colð� 0:43Þ, as seen in Figs. 4b and 5a, and egðDT �colÞ ¼0, resulting in columnar growth for condition B in Fig. 3a.

M.A. Martorano, V.B. Biscuola / Acta Materialia 57 (2009) 607–615 613

On the other hand, when DT �r increases from 0.01 to 1,the undercooling distribution widens (Fig. 5a), enablingnucleation of equiaxed grains below DT �col. Accordingly,for V* < 1 the boundary curve in Fig. 3a shifts to the right,favoring equiaxed solidification. When DT �r is furtherincreased from 1 to 10, the same behavior encountered incondition A is recovered (equiaxed growth is less likelybecause some area under the undercooling distributioncurve is removed by the liquidus and columnar front iso-therms) and columnar solidification is favored.

To sum up, two types of behavior exist regarding theeffect of DT �r on the growth maps. When V* > 1, the aver-age nucleation undercooling for equiaxed grains (DT* = 1)is smaller than the columnar front undercooling (i.e.DT �col > 1 in Fig. 3a). Therefore, for instantaneous nucle-ation (DT �r ¼ 0:01), all possible equiaxed grains nucleate,representing the most likely condition for equiaxed growth.For an increase in DT �r above �1/3 or above� ðDT �col � 1Þ=3, some area under the curve of nucleationundercooling distribution is cut off, decreasing the numberof nucleated equiaxed grains and widening the columnarsolidification field. When V* < 1, the average nucleationundercooling for equiaxed grains is now larger than thecolumnar front undercooling (i.e. DT �col < 1 in Fig. 3a).Then, for instantaneous nucleation, no equiaxed grainsnucleate and only columnar growth occurs for any temper-ature gradient. Increasing DT �r causes some area under thecurve of nucleation undercooling distribution to lie belowDT �col, enabling some nucleation of equiaxed grains andreducing the columnar field. For a larger DT �r increase(DT �ra1Þ, however, the columnar solidification field widensagain because the undercooling distribution curve is nowcut off by the liquidus isotherm and less equiaxed grainnucleation occurs.

A similar behavior is observed in the effects of r/ for alog-normal distribution of nucleation undercoolings(Fig. 3b). Differences in the shift of the boundary curvesoccur, however, because the liquidus isotherm never cutsoff the nucleation distribution curves (Fig. 5b).

4. Transient unidirectional solidification

In the previous section, modifications in the growthmaps were explained for a change in the spread of the dis-tribution of nucleation undercoolings. Since experimentalvalidation of this theory is difficult, an attempt is made tovalidate it using macrostructures calculated with a sto-chastic model for the transient unidirectional solidificationof an Al–7 wt.% Si alloy. The stochastic model imple-mented and used in the present work is based on themodel proposed by Gandin and Rappaz [21] andconsists of a transient one-dimensional macroscopicmodel and a two-dimensional microscopic model of cellu-lar automaton. To verify its correct implementation, somemodel results were compared with those presented byRappaz and Gandin [28,29], and showed excellentagreement.

In the macroscopic model, the transient heat conductionequation was solved in a one-dimensional domain 0.15 mlong. One of the domain boundaries was adiabatic, whereasthe heat flux out of the other boundary was q ¼ hðT � T1Þ,where h is the heat transfer coefficient (250 W m�2 K�1), andT1 is a reference temperature (298 K). A uniform tempera-ture (991 K) was adopted throughout the domain as theinitial condition.

In all simulations, the properties for the Al–7 wt.% Sialloy were: js = 137.5 W m�1 K�1 (heat conductivity ofsolid); jl = 60.5 W m�1 K�1 (heat conductivity of liquid);L = 387.4 � 103 J kg�1 (latent heat of fusion); cP =1126 J kg�1 K�1 (specific heat); q = 2452 kg m�3 (density);k = 0.13 (solute partition coefficient); Tf = 933 K (meltingpoint of pure Al); TL = 891 K (liquidus temperature);and TE = 850 K (eutectic temperature).

In the two-dimensional microscopic model, the nucle-ation of grains was simulated with the normal distributionof nucleation undercoolings described by Eq. (1). Two setsof distribution parameters were used: one for the bulkliquid and another for the liquid layer adjacent to theboundary through which heat was extracted. For the liquidlayer, DT N ;layer ¼ DT r;layer ¼ 0 (instantaneous nucleation)and nT,layer = 3 � 106 m�2, which was converted for atwo-dimensional simulation domain using the relationn2D

T ;layer ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinT ;layer=p

p[28]. For the bulk liquid, DT N ¼ 2

or 5 K, and DTr was adjusted to give 0 6 DT �r 6 3. Thenumber density of grains was nT = 5 � 106 m�3, whichwas converted for the two-dimensional domain usingn2D

T ¼ ðnT

ffiffiffiffiffiffiffiffi6=p

pÞ2=3 [28]. To simulate grain growth, the

velocity of the grain envelope diagonals was calculatedusing the same kinetic equation as that used to derive thepresent model equations (Eq. (5)), with constantsA = 3 � 10�6 m s�1 K�2.7 and m = 2.7.

In the macroscopic model, the transient heat conductionequation was solved numerically by the explicit finite-vol-ume method [30] using a one-dimensional mesh of 30 equalvolumes. In the microscopic model, a two-dimensional cel-lular automaton mesh of 300 (heat flow direction) � 100(perpendicular to heat flow) square cells was adopted. Fur-ther details of the stochastic model can be found elsewhere[21].

Calculated grain macrostructures as a function of DT �rare presented for DT N ¼ 2 K (Fig. 6a) and 5 K (Fig. 6b).The CET region was defined as that where the aspect ratioof grains was between 0.3 and 0.4, as explained by Biscuolaand Martorano [26]. As can be seen, the CET positionmight change significantly for constant DT N when DT �r isincreased, showing the importance of DT �r. Furthermore,the CET frequently predicted with deterministic modelsfor instantaneous nucleation (DT �r ¼ 0) can be remarkablydifferent from that obtained for a distribution ofundercoolings.

In Fig. 7, solidification paths, defined as the dimension-less columnar front velocity as a function of the dimension-less temperature gradient at the front during solidification,are superimposed on the growth map of Fig. 3a. The front

Fig. 6. Macrostructures of an Al–7 wt.% Si alloy calculated with the stochastic model as function of DT �r for DT N : (a) 2 K; (b) 5 K. The CET region isbetween the two horizontal lines.

Fig. 7. Solidification paths for the simulations in Fig. 6 with DT N ¼ 2and 5 K, superimposed on the growth map of Fig. 3a.

614 M.A. Martorano, V.B. Biscuola / Acta Materialia 57 (2009) 607–615

velocity was calculated from successive front positions,determined as the location of the first solid cell observedfrom the bulk liquid towards the solid. Note that the

solidification paths, calculated with a transient model, areplotted on growth maps based on a steady-state model.This implies that quasi-steady-state is assumed at thecolumnar front of the stochastic model. The quasi-steady-state assumption has successfully enabled steady-state models of dendritic [27] and nondendritic [13] growthto be extensively used in transient models of solidification[13,21]. Furthermore, in the stochastic model the columnarfront blocking has been observed to occur according to themechanical blocking criterion [26], as assumed to constructthe growth maps.

Numerous simulations with the stochastic model indi-cated that the solidification paths are virtually unchangedwith DT �r. Nevertheless, the length of each path, whichended at the CET, was different. Therefore, only the longestpath is shown for each series of simulations of Fig. 6. In bothcases, G* is initially constant and V* increases abruptly,reaching a constant value. Afterwards, G* begins to decreaseuntil the CET occurs.

The solidification path for DT N ¼ 2 K intercepts theboundary curves at a region where V* > 1 (DT �col > 1). Asdiscussed previously, in this case an increase in DT �rdecreases the number of nucleated equiaxed grains, favor-ing columnar growth. This is observed in the macrostruc-

M.A. Martorano, V.B. Biscuola / Acta Materialia 57 (2009) 607–615 615

tures of Fig. 6a where the columnar region increases, rais-ing the CET position, and the number of equiaxed grainsdecreases, increasing grain size.

For DT N ¼ 5 K, the solidification path intercepts theboundary curves at V* < 1 (DT �col < 1) in Fig. 7. For instan-taneous nucleation (DT �r ¼ 0:01), equiaxed growth and theCET are not observed in Fig. 6b, because the solidificationpath does not intercept the boundary curve in Fig. 7. AsDT �r increases from 0.01 to 0.1 or 0.3, the path now inter-cepts the boundary curves, since the spread of the nucle-ation undercooling distribution increases and equiaxedgrains begin to nucleate, causing a CET. A further increasein DT �r from 0.3 to 2 or 3 decreases again the number ofnucleated equiaxed grains as explained before for V* < 1,increasing the size of the columnar region.

5. Concluding remarks

A mathematical model for unidirectional steady-statesolidification is proposed to predict the CET when equiaxedgrains nucleate according to either a normal or a log-normaldistribution of nucleation undercoolings. The present modelreduces to Hunt’s [3] model when the spread of the distribu-tion decreases, approaching instantaneous nucleation.Growth maps of dimensionless isotherm velocity (V*) andcolumnar front undercooling (DT �col) vs. dimensionless tem-perature gradient are constructed. The maps are affected bythe spread of the distribution of nucleation undercoolings intwo different ways. When V* > 1 (DT �col > 1), increasing thespread results in less favorable conditions for equiaxedgrowth, widening the field of columnar solidification. Other-wise (V* < 1 and DT �col < 1), there are only columnar grainsfor a vanishing distribution spread (instantaneous nucle-ation), but equiaxed growth is favored when the spreadincreases, narrowing the field of columnar solidification.For a larger spread, however, equiaxed grain nucleation ishindered, widening the columnar solidification field again.This behavior was confirmed in the grain macrostructuresof a transient stochastic model. An increase in the distribu-tion spread changed a completely columnar grain structureinto one with �50% of equiaxed grains. Therefore, predic-tions of the CET available in the literature and obtained withmodels that assume instantaneous nucleation conditionsshould be considered carefully.

Acknowledgments

The authors thank Fundac�ao de Amparo a Pesquisa doEstado de Sao Paulo (FAPESP) for the financial support(Grant 03/08576-7) and Conselho Nacional de Desenvolvi-mento Cientıfico e Tecnologico (CNPq) (Grant 475451/2004-0) for the scholarship to V.B.B.

References

[1] Flemings MC. Solidification processing. New York: McGraw-Hill;1974.

[2] Irving WR. Continuous casting of steel. London: Institute ofMaterials; 1993.

[3] Hunt JD. Mater Sci Eng 1984;65:75.[4] Nguyen-Thi H, Reinhart G, Mangelinck-Noel N, Jung H, Billia B,

Schenk T, et al. Metall Mater Trans A 2007;38A:1458.[5] Spittle JA. Int Mater Rev 2006;51:247.[6] Martorano MA, Beckermann C, Gandin CA. Metall Mater Trans A

2003;34A:1657.[7] Banaszek J, McFadden S, Browne DJ, Sturz L, Zimmermann G.

Metall Mater Trans A 2007;38A:1476.[8] Mc Fadden S, Browne DJ. Scripta Mater 2006;55:847.[9] Wu M, Ludwig A. Metall Mater Trans A 2007;38A:1465.

[10] Sturz L, Drevermann A, Pickmann C, Zimmermann G. Mater SciEng A 2005;413:379.

[11] Martorano MA, Capocchi JDT. Int J Cast Metals Res 2000;13:49.[12] Thevoz P, Desbiolles JL, Rappaz M. Metall Trans A 1989;20:311.[13] Greer AL, Bunn AM, Tronche A, Evans PV, Bristow DJ. Acta Mater

2000;48:2823.[14] Quested TE, Greer AL. Acta Mater 2005;53:4643.[15] Wang CY, Beckermann C. Metall Mater Trans A 1994;25:1081.[16] Flood SC, Hunt JD. J Cryst Growth 1987;82:552.[17] Jacot A, Maijer D, Cockcroft S. Metall Mater Trans A 2000;31:2059.[18] Martorano MA, Biscuola VB. Modell Simul Mater Sci Eng

2006;14:1225.[19] Quested TE, Greer AL, Cooper PS. Mater Sci Forum 2002;396(4):53.[20] Quested TE, Greer AL. Acta Mater 2004;52:3859.[21] Gandin CA, Rappaz M. Acta Mater 1997;45:2187.[22] Dong HB, Lee PD. Acta Mater 2005;53:659.[23] Cho SH, Okane T, Umeda T. Int J Cast Metal Res 2001;13:327.[24] Ares AE, Schvezov CE. Metall Mater Trans A 2000;31:1611.[25] Siqueira CA, Cheung N, Garcia A. J Alloy Compd 2003;351:126.[26] Biscuola VB, Martorano MA. Metall Mater Trans A, in press.

doi:10.1007/s11661-008-9643-x.[27] Lipton J, Glicksman ME, Kurz W. Mater Sci Eng 1984;65:57.[28] Rappaz M, Gandin CA. Acta Metall Mater 1993;41:345.[29] Gandin CA, Rappaz M. Acta Metall Mater 1994;42:2233.[30] Patankar SV. Numerical heat transfer and fluid flow. New

York: Hemisphere; 1980.