FREE GROWTH OF EQUIAXED CRYSTALS SETTLING IN

UNDERCOOLED NH4Cl±H2O MELTS

B. APPOLAIRE, V. ALBERT, H. COMBEAU and G. LESOULT{LSG2M, Ecole des Mines, F-54042 Nancy Cedex, France

(Received 9 January 1998; accepted 7 June 1998)

AbstractÐRecently theoretical works concerning the e�ect of convection on the growth of isolated den-drites have been compared with experiments on NH4Cl settling equiaxed crystals. It was inferred that moreaccurate theories were still needed to describe properly the equiaxed crystal growth in the presence of con-vection. Some new results have been obtained using the following experimental set-up: in a tube containingan undercooled solution of NH4Cl±H2O, settling NH4Cl equiaxed crystals have been ®lmed with a videocamera so as to determine the evolution with time of their size and of their settling velocity. After a carefulcomparison of the experimental results with some calculations involving the choice of a stability constant,no major discrepancy has been found to prevent the application of the theories in question to movingequiaxed crystals. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.

Re sumeÂÐRe cemment, certains travaux the oriques e tudiant l'e�et de la convection sur la croissance desdendrites isole es ont e te confronte s aÁ des expe riences mettant en jeu des cristaux e quiaxes de NH4Cl se di-mentant. Les modeÁ les the oriques apparurent alors trop impre cis pour de crire correctement la croissancedes cristaux e quiaxes en pre sence de convection. Dans notre e tude sont pre sente s de nouveaux re sultatsexpe rimentaux obtenus avec le dispositif suivant: dans un tube contenant un me lange NH4Cl±H2O liquideen surfusion, nous avons ®lme des cristaux e quiaxes de NH4Cl se dimentant a®n de de terminer l'e volutionde leur taille et de leur vitesse de chute au cours du temps. A la suite de comparaisons soigneuses entre cesre sultats expe rimentaux et des conside rations the oriques mettant en jeu le choix d'une constante de stabi-lite , nous n'avons pas trouve de di�e rences majeures qui interdiraient l'application des the ories en questionau cas des cristaux e quiaxes en mouvement. # 1998 Acta Metallurgica Inc. Published by Elsevier ScienceLtd. All rights reserved.

1. INTRODUCTION

In recent years, the prediction of the evolution of

equiaxed crystals moving in a solidifying melt has

appeared as a crucial point for the modelling of

solidi®cation processes concerning the metallurgical

industry. The relative solid/liquid movement

changes the ¯uid ¯ow, and the transport of heat,

mass and solute over the whole range of scales

involved in the solidi®cation. Therefore, it has an

important e�ect on the quality of the as-cast pro-

ducts. First, attention has been focused on the

growth kinetics of the tip of an isolated dendrite in

the presence of an axisymmetric ¯ow of melt.

Experimental studies as well as theoretical ones

have shown good agreement concerning the e�ect

of such a ¯ow on the growth of a single paraboloi-

dal crystal from the melt [1±5]. Recently, a model

has been proposed for describing the evolution of a

whole three-dimensional equiaxed crystal which ten-

tatively takes into account the e�ects of the convec-

tion using some previous results on the growth

kinetics of the dendrite tips [6]. In fact, this model

is based on the theory of the coupled transports of

mass, solute and momentum obtained by Ananth

and Gill [3] together with a result from the micro-

scopic solvability theory valid for purely di�usive

growth [7, 8]. In order to check the validity of such

a model in the case of equiaxed crystals, Ramani

and Beckermann [9] carried out experiments on

equiaxed dendritic crystals settling in transparent

NH4Cl±H2O melts. By these means they were able

to study the e�ect of a relative melt ¯ow with

respect to the crystals on the growth velocity of

these crystals. So as to compare quantitatively the

calculations and the experiments, they chose a value

of the stability constant s* very close to that evalu-

ated experimentally by Liu et al. [10] in the case of

purely di�usive growth. They concluded there was

lack of accuracy of the theories involved in the

model which seemed inappropriate for describing

the growth of a whole equiaxed crystal. Here, new

experimental results are presented dealing with the

growth velocity of NH4Cl settling crystals. They are

compared to the same model of dendrite tip growth

which accounts for the e�ect of ¯uid ¯ow used by

Ramani and Beckermann [9]. However, contrary to

the choices of Ramani and Beckermann, we take

into account the e�ect of the intensity of the ¯uid

¯ow on the actual value of the stability constant s*

Acta mater. Vol. 46, No. 16, pp. 5851±5862, 1998# 1998 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reservedPrinted in Great Britain

1359-6454/98 $19.00+0.00PII: S1359-6454(98)00236-5

{To whom all correspondence should be addressed.

5851

which allows an estimation of the stability constantthat is good for ¯ow velocities of the order of0.01 m/s. Finally it is concluded that it is not poss-

ible with the present results to decide whether theavailable theories are accurate but they can berather useful.

2. EXPERIMENTS [11]

2.1. Experimental procedure

Experiments were performed with NH4Cl±H2O

melts in a glass tube (1.5 m in height, 0.04 m ininner diameter) mounted in a vertical plastic box(1.45 m in height, 0.1� 0.1 m2 in section). Water

from a thermostat regulated with an accuracy of0.18C circulated in the box so as to set the tempera-ture of the melt which can reach a stationary valueafter 30 min at the latest. A maximum temperature

di�erence of ÿ0.258C was found between the waterand the melt due to heat exchanges with the roomair at the top of the glass tube. Thanks to a travel-

ling video camera, the position, the orientation, theshape and the size of selected crystals were continu-ously ®lmed during their evolution from the top to

the bottom of the tube. A schematic drawing of theexperimental set-up is shown in Fig. 1.At the beginning of each experiment a few grains

of NH4Cl powder (50±200 mm in size) were injectedat the top of the glass tube. Two compositions werechosen for the melt: 34 and 30% of NH4Cl. Thee�ect of undercooling was studied for the 34%

NH4Cl±H2O mixture (DT = 1 or 38C). The e�ectof the composition was studied for the highestundercooling of 38C (34 or 30% of NH4Cl). In

each case four or ®ve runs were selected duringwhich the behaviour of one crystal was studied.One must note that for the experiment with a 30%

NH4Cl±H2O composition at 38C of undercoolingtwo di�erent melts were prepared. The liquidustemperature of each melt was measured by thermal

analysis before the growth experiments so that theactual undercooling was known with an accuracy of[ÿ0.68C; +0.68C]. The experimental conditions for

the present results are reported in Table 1 with therelated supersaturations [see equation (2) in Section3.1 for the de®nition of the supersaturation O]. Thesettling and growing crystals were ®lmed when the

crystals reached the zone of homogeneous tempera-ture, after a 10 cm fall. In fact, each crystal was fol-lowed for more than 60 s (sometimes up to 100 s)

on a vertical distance of more than 50 cm (some-times up to 80 cm). Then the video recordings wereused to assess the size of the crystals (i.e. the appar-

ent distance between the tips of two opposite pri-mary dendrite arms) and their vertical position vstime.

2.2. Experimental observations and results

Two types of situations can be distinguished

depending on whether the crystals are spinning ornot: the crystals which are spinning in the ¯owshow a symmetric morphology [Fig. 2(a)], whereasthose which keep their relative position towards the

Fig. 1. Schematic drawing of the experimental set-up.

APPOLAIRE et al.: FREE GROWTH OF EQUIAXED CRYSTALS5852

¯ow become asymmetric [Fig. 2(b)]. In the last case,

the dendrite tips of the crystal which are exposed to

convection grow faster than the others.

Figure 3 shows an example of the results: the

height and size vs time of a crystal growing as it

settles in the undercooled tube. The recordings of

the height vs time are accurate and very well ®tted

with second-order polynomials, as shown in Fig. 3for one crystal falling in a 34% NH4Cl±H2O alloy

at 38C of undercooling. Thus it is possible to calcu-

late the instantaneous settling velocity quite accu-

rately and to show its linear dependence on time.

Actually the crystals fall with an increasing velocity

as they settle in the tube. The recording of the size

of the crystals vs time is much more scatteredbecause of the resolution of the video camera

(0.2 mm) and because of possible spinning move-

ments of the crystals during their fall. In fact, when

the dendrite stems are not perpendicular to the opti-

cal axis of the video camera, the apparent size of

the crystals is smaller than the actual one. The sche-

matic drawing in Fig. 4 shows that the actual sizecan be as large as Z2 times the apparent one.

Therefore, the uncertainty interval for the size was

estimated as follows: [apparent sizeÿ 0.2 mm;

Z2�apparent size + 0.2 mm]. For the smallest sizes

which are of the order of the optical resolution

limit, the relative uncertainty reaches about 100%.

Hence, it is not possible to calculate instantaneous

growth velocities from the previous measurements

without some preliminary data treatment. However,

it is possible to calculate values of the growth vel-

ocity averaged over the total time of the fall, bydividing the di�erence between the last and ®rst

measured values of the crystal size by the total

settling time. In a similar way, average settling vel-

Table 1. Experimental conditions and order of magnitude of growth velocities

Melt composition (wt%) Undercooling (8C) SupersaturationsActual growthvelocities (m/s)

Theoretical purely di�usive growthvelocities (m/s)

34 1 3.15� 10ÿ3 15� 10ÿ6 7.4�10ÿ8

34 3 9.38� 10ÿ3 12.5� 10ÿ5±3� 10ÿ5

9.7�10ÿ7

30 3 8.85� 10ÿ3 11.5� 10ÿ5±2.5� 10ÿ5

9.1�10ÿ7

Fig. 2. Evolution of (a) symmetric and (b) asymmetric settling crystals after 0.2 and 0.8 m fall (case30% NH4Cl±H2O alloy, undercooling 38C); approximate scale 3:1.

APPOLAIRE et al.: FREE GROWTH OF EQUIAXED CRYSTALS 5853

ocities can be obtained by dividing the ®rst and last

measured values of the position by the total settling

time. For each crystal the average growth velocity

has been reported vs the related average settling vel-

ocity in Fig. 5. As will be discussed later, the scatter

of the points which correspond to the average

growth velocities for the 30% NH4Cl±H2O alloy at38C of undercooling might be due to the uncer-

tainty on the undercooling measurement.

Furthermore, for a given melt, the scatter of theaverage growth velocity is of the order of the uncer-

tainty on the size measurement.

Two de®nite conclusions can be drawn from

these experimental results. The ®rst one is theincrease of growth velocity with undercooling

(Fig. 5). It is a well-known tendency for a purely

di�usive dendrite growth [12]. The second con-clusion is related to the e�ect of the settling velocity

of the equiaxed crystals on their growth velocity:the larger the settling velocity, the faster the den-

drite growth. This phenomenon can be observed

during the fall of an isolated crystal, as in Fig. 3:the slight trend of the experimental curve of the size

vs time to an upward concavity can be attributed to

the instantaneous acceleration of the dendritegrowth which results from the instantaneous accel-

eration of the crystal fall as noticed before. From

Fig. 5 the same conclusion can be drawn indepen-dently: when comparing di�erent crystals growing

in the same 34% NH4Cl±H2O alloy at the same

undercooling of 38C, it seems that the crystalswhich fall more rapidly (because their initial size is

likely slightly larger) are growing faster. Lastly, inFig. 5, one must note a slight shift between the sets

of plots corresponding to 30% NH4Cl±H2O and

34% NH4Cl±H2O alloys at the same undercoolingof 38C which might be due to the di�erence in the

composition. As it will be shown brie¯y in the next

section these tendencies are consistent with predic-tions of some existing theoretical models.

Fig. 3. Position and size vs time of a crystal settling in a 34% NH4Cl±H2O alloy at 38C of undercool-ing. On the position curve, crosses represent the experimental data when the continuous line represents

the best ®t with second-order polynomial interpolation.

Fig. 4. E�ect of the spinning movement of the equiaxedcrystals on the size measurement.

APPOLAIRE et al.: FREE GROWTH OF EQUIAXED CRYSTALS5854

3. PREDICTED GROWTH VELOCITY

3.1. Description of the model

The problem of dendrite growth in the case ofpurely di�usive transport (transport of heat forpure materials and transport of solute for alloys)

has been extensively studied since 1947 [12, 13].Ivantsov solved the problem of heat transport andderived a relationship between the undercooling of

the melt and the thermal Pe clet number of a para-boloidal and isothermal dendrite tip growing at aconstant velocity in an in®nite pure medium when

the overall kinetics is controlled by the heat conduc-tion in the liquid. This has been extended to thecase of the growth of a paraboloidal dendrite tip ina binary melt when the overall kinetics is controlled

by solute di�usion in the liquid. Figure 6 illustratesschematically the temperature and concentration®elds around an equiaxed dendritic crystal growing

in such conditions. The temperature of the dendritetips is almost equal to that of the bulk liquidwhereas a de®nite concentration gradient exists

between the liquid near the tips and the bulk.As long as the temperature can be considered

homogeneous in the liquid during the growth, it is

possible to de®ne a supersaturation which drivesthe growth and to relate it to a chemical Pe cletnumber as follows [14]:

O � Pec � exp�Pec� � E1�Pec� �1�where O is the supersaturation, Pec the chemicaldi�usion Pe clet number related to the tip growthvelocity, and E1 the exponential integral.

Fig. 5. Average growth velocity vs average settling velocity for each crystal.

Fig. 6. (a) Schematic temperature pro®le around a growingequiaxed crystal. (b) Schematic related concentration pro-®le. (c) Schematic phase diagram of the binary alloy under

study (see text for de®nitions).

APPOLAIRE et al.: FREE GROWTH OF EQUIAXED CRYSTALS 5855

The supersaturation is de®ned for the tempera-

ture imposed in the liquid far from the dendrite. It

is related to the undercooling which can be de®ned

with respect to the liquidus temperature TL associ-

ated with the composition of the liquid far from the

dendrite as follows:

O � w* ÿ w1

w* � �1ÿ k� �DT

�1ÿ k� � �DTÿm � w1� �2�

with w* and w1 the mass fractions of solute in the

liquid at the solid/liquid interface and in the bulk,

respectively, k the partition coe�cient, m the slope

of the liquidus line, and DT the undercooling of the

bulk liquid de®ned as follows:

DT � TL�w1� ÿ T 1 � TL�w1� ÿ TL�w*�:

Finally, the chemical di�usion Pe clet number Pecis expressed as

Pec � rt � vt2Dl

�3�

with vt the tip velocity, rt the tip radius, and Dl the

solute di�usivity in the liquid.

But taking the transport phenomena into account

is not su�cient to determine a single solution. In

fact, the selection of the dendrite tip is related to

anisotropic properties of the interface such as the

interfacial energy or the molecular attachment kin-

etic constant. For NH4Cl±H2O alloys, it is not clear

whether one of these properties is predominant over

the other. On the one hand, the experimental works

of Raz et al. [15] and Tanaka and Sano [16] refer to

the molecular attachment kinetics to explain the ex-

perimental values of the interfacial concentration

they measured in the liquid. On the other hand,

Chan et al. [17] and Liu et al. [10] report on exper-

imental results for which the e�ect of interfacial

energy seems to dominate. In particular, Liu et al.

have analysed their data thanks to the microscopic

solvability theory [7, 8] which predicts a constancy

of the product �vt � r2t � for a given supersaturation:

vt � r2t �Dl � d0s*

�4�

where s* is the stability constant dependent on the

physical properties of the alloy [7] as well as on its

composition [18] and where the capillarity length d0is related to the Gibbs±Thompson constant G as

d0 � G

j m j �w* � �1ÿ k� : �5�

Finally, simple combinations of equations (3)±(5)

lead to the following expression of the growth vel-

ocity:

vt � Dl � s*

G� 4� j m j �w* � �1ÿ k� � Pe2c : �6�

The agreement between the experiments and the

theory was found by Liu et al. [10] to be quite good

in the particular case of a 31% NH4Cl±H2O alloy:

it leads to a relevant estimate of the stability con-

stant valid for the binary NH4Cl±H2O mixtures

under study, i.e. s* � 0:026. Here, the results of

Liu et al. [10] have been chosen as a basis for esti-

mating the stability constant when the convection

can be neglected, assuming that the e�ect of the

melt composition on the stability constant was

weak in the range of composition (30±34% of

NH4Cl). The values of the predicted growth rate in

a purely di�usive regime have been calculated

thanks to equations (1), (2) and (6) and reported in

Table 1. They are much smaller than the actual

growth velocities as measured here.

The study of the problem of dendrite growth in a

¯owing melt is more recent and still more contro-

versial than the purely di�usive case. Concerning

the result of the microscopic solvability theory, sev-

eral authors think that equation (6) holds whether

the melt is at rest or not. It is also agreed that the

stability constant s* should depend on the melt vel-

ocity, however, it is not clear how. The microscopic

solvability theory predicts a decrease of s* with an

increasing melt velocity [4], when some experiments

performed by Lee et al. on succinonitrile show the

opposite tendency [1]. Therefore, it is proposed here

to choose the value of the stability constant s* for

a good quantitative ®t between the predictions of

the theory and the experimental results after a care-

ful choice of the values of the materials data

(Section 3.2).

Concerning the solute mass transport, it is necess-

ary to revisit equation (1) relating the supersatura-

tion to the chemical di�usion Pe clet number. It is

possible to use the theoretical results which have

been established by Ananth and Gill [3] in the case

of purely thermal dendrite growth thanks to the

similarity of the mathematical equations which

describe the heat transfer around a dendrite tip

growing in a pure melt on the one hand, and the

solute mass transfer around a dendrite tip growing

in an isothermal binary liquid mixture on the other

hand. In fact, Ananth and Gill solved the heat

transport problem in the melt around an isolated

dendrite tip, when the ¯ow is axisymmetric in the

opposite direction to the dendrite growth for some

solutions of the Navier±Stokes equation in speci®c

cases (e.g. Stokes ¯ow, Oseen viscous ¯ow and po-

tential ¯ow). They obtained general relations which

have been transposed here as follows:

O � F�Pec,Peu� �7�where F is a function reducing to the so-called

Ivantsov solution [equation (1)] when the melt is at

rest, and where Peu is the chemical Pe clet number

related to the velocity of the melt with respect to

the tip U1, i.e.

APPOLAIRE et al.: FREE GROWTH OF EQUIAXED CRYSTALS5856

Peu � rt �U 12 �Dl

:

For a given supersaturation O, the function F is

such that the growth Pe clet number Pec increases

when the melt ¯ow Pe clet number Peu increases, i.e.

usually when the velocity U1 increases, in the

opposite direction to the growth.

The choice of an approximate solution depends

on the value of the Reynolds number related to the

dendrite tip:

Re � U 1 � rt�

where n is the kinematic viscosity of the melt.

The so-called Stokes ¯ow and Oseen viscous ¯ow

solutions hold for small Reynolds numbers whereas

the so-called potential ¯ow solution is valid for

large ones. As long as the supersaturation can be

taken as an upper value for the Pe clet number [this

crude approximation holds for the so-called

Ivantsov solution equation (1) when the Pe clet

number Pec is below around 0.1], it is possible to

estimate an upper limit to the Reynolds number as

follows:

Re<2 � O �U1

vt�D

l

�: �8�

From the experimental conditions and measure-

ments, it then results that the Reynolds number

should be smaller than 0.01. Consequently, in this

study, the most appropriate approximation pro-

posed by Ananth and Gill should be the Stokes

¯ow solution, provided that the supersaturation is

larger than the chemical Pe clet number and know-

ing that the approximation of Oseen viscous ¯ow

gives similar results for a Reynolds number smaller

than unity, with a more complicated function F. In

fact, the range of Reynolds numbers covered by the

calculations has been checked a posteriori: it

appears that its upper limit is about 0.07, a value

which con®rms the validity of the Stokes ¯ow sol-ution in the present study.

3.2. Choice of materials data

In Table 2 are reported the chosen values of thedata related to the phase diagram which are thoseused by Ramani and Beckermann [9] and the values

of viscosity and density of the melt which are thoseused by Jang and Hellawell [21].Special attention was paid to the choice of the

values of the mass di�usion coe�cient Dl and ofthe Gibbs±Thompson constant G. In fact, since itwas chosen here to ®t the value of the stability con-

stant s*, the proper quantity, the value of whichhas to be checked, is the ratio Dl/G to which the tipvelocity is proportional [equation (6)]. Data havebeen collected from di�erent sources and compared

in Table 3. The values of G reported in the ®rst twocolumns of Table 3 have been calculated on thebasis of the values of the capillary length d0 that

have been estimated by Liu et al. [10] and Tanakaand Sano [16] from their experimental measure-ments, using Dougherty and Gollub's

procedure [19]. In fact, equation (5) allows the esti-mate of G from d0 provided that the local compo-sition w* is approximately equal to the nominal one

w1. The di�erence between their two estimates of Gresults from the choice of the mass di�usion coe�-cient which plays a role in the experimental evalu-ation of d0. The values of Dl and d0 published by

Liu et al. may indeed appear quite atypical relativeto other sources. However, the related value of theratio Dl/G is of the same order of the other esti-

mates of this important quantity, except Ramaniand Beckermann's estimate [9] (Table 3). Finally,among the couples of values which were proposed

for the mass di�usion coe�cient Dl and the Gibbs±Thompson constant G and which lead to the mostfrequently quoted order of magnitude of the ratioDl/G, the values selected by Blackmore et al. [20]

have been chosen in the present study.

3.3. Choice of the stability constant

Equation (7) with Stokes ¯ow approximation andequation (6) have been solved for the measured

undercoolings and a large range of settling vel-ocities. In order to match the theoretical tip vel-ocities thus obtained to the average experimental

ones within the explored domains of the settling vel-ocities, the value of s* has been adjusted relative to

Table 2. Values of quantities used in the calculations

Quantity Value References

Liquidus slope, m ÿ4.8 K/wt% [9]Partition coe�cient, k 0 [9]Dynamic viscosity, m 1.03� 10ÿ3 Pa s [21]Density of the liquid, rl 1080 kg/m3 [21]Mass fraction of water inthe liquid mixture, w1

66, 70 and 72 wt%

Table 3. Values of the solute di�usion coe�cient Dl and of the Gibbs±Thompson constant G for NH4Cl±H2O alloys found in theliterature

Tanaka and Sano [16] Liu et al. [10] Blackmore et al. [20] Ramani and Beckermann [9]

Dl (m2/s) 2.6� 10ÿ9 5� 10ÿ10 2.3�10ÿ9 2� 10ÿ9

G (m K) 5.09� 10ÿ7$ 6.6� 10ÿ8$ 3.54�10ÿ7 [5�10ÿ9±4� 10ÿ8]d0=15.9� 10ÿ10 m d0=2� 10ÿ10 m

Dl/G (m/K s) 5.1� 10ÿ3 7.58� 10ÿ3 6.5�10ÿ3 [5�10ÿ2±0.4]

$From the experimental evaluation of the capillary length d0.

APPOLAIRE et al.: FREE GROWTH OF EQUIAXED CRYSTALS 5857

Fig. 7(a), (b) and (c)ÐCaption opposite.

the experimental results for the 34% NH4Cl±H2O

alloy at 18C of undercooling [Fig. 7(a)]. The choiceof this set of experimental results has been decidedbecause the scatter of the average growth velocities

appears to be less important than for the two othersets (Fig. 5). The value of s* which corresponds tothe best ®t between the mean experimental plots

and the theoretical curves in Fig. 7(a) is about0.081. Since this value is signi®cantly di�erent fromthe value published by Liu et al. [10] in the purelydi�usive case, it is proposed here to account for a

possible in¯uence of the ¯uid ¯ow on the dendritegrowth in the ®nal choice of the s* value for thetheoretical predictions. For small settling velocities,

it was chosen to keep the value given by Liu et al.,i.e. s* � 0:026. For settling velocities of the sameorder of the average experimental settling velocities,

i.e. 0.01 m/s, it was proposed to carry out calcu-lations with the ®tting value: s* � 0:081.

3.4. Predicted results

Two types of predictions were conducted: on theone hand, the assessment of the e�ect of the settling

velocity on the dendrite growth for given under-coolings; on the other hand, the simulation of thetime dependent growth of some isolated dendritic

crystals settling in an undercooled melt.The dendrite tip velocity has been calculated for

a few given melt compositions and undercoolings as

a function of the intensity of a hypothetical axisym-metrical liquid ¯ow around the tip. First, calcu-lations were carried out using the s* value foundby Liu et al. [7] in the purely di�usive case, i.e.

s* � 0:026. The related curves are plotted in

Fig. 7(a)±(c) in interrupted lines. They illustrate the

large predicted e�ect of the ¯uid ¯ow on the den-

drite growth when the relative liquid/crystal vel-

ocities are small. The present experimental data arereported in the same graphics, assuming that the

velocity of the liquid ¯ow around the tips of the

equiaxed dendritic crystals falling in the tube is

equal to the average settling velocity. The exper-

imental points are systematically above the former

theoretical curves. Since it is proposed here toaccount for a possible in¯uence of the ¯uid ¯ow on

the value of the stability constant s*, the portion of

the theoretical curves calculated with the ®tting

value of s*, i.e. s* � 0:081 is also plotted in

Fig. 7(a)±(c) in the velocity range 0.006±0.011 m/s.

The ®t is reasonably good for all the present results.Finally, the experimental results published by

Ramani and Beckermann [9] are plotted in Fig. 7(d)

together with the related curves calculated with the

same present ®tting s* value. The growth velocities

reported by these authors are systematically larger

than those which are predicted following the pre-sent procedure.

Secondly, a few simulations of the time evolution

of the size of settling crystals have been plotted inFig. 8 for comparison with experimental obser-

vations. As it has been mentioned, for each selected

experimental run, the evolution with time of the

vertical position can be approximated by second-

order polynomials (Fig. 3), so as to deduce instan-taneous settling velocities. Then, equations (7) and

(6) have been used to calculate the instantaneous

Fig. 7. Growth velocity vs settling velocity for: (a) 34% NH4Cl±H2O and DT = 18C; (b) 34% NH4Cl±H2O and DT = 38C; (c) 30% NH4Cl±H2O and DT= 38C; (d) 28% NH4Cl±H2O and several

undercoolings [9].

APPOLAIRE et al.: FREE GROWTH OF EQUIAXED CRYSTALS 5859

Fig. 8. Size vs time for a crystal: (a) in a 34% NH4Cl±H2O alloy at 18C of undercooling; (b) in a 34%NH4Cl±H2O alloy at 38C of undercooling; (c) in a 30% NH4Cl±H2O alloy at 38C of undercooling.

dendrite tip velocity knowing the measured under-cooling, the instantaneous value of the settling vel-

ocity and the previous ®tting value of s*. Finally,the calculated instantaneous growth velocity hasbeen integrated with time from the initial measured

sizes to generate a simulated curve of the size vstime. The comparisons between some simulationsand observations are shown in Fig. 8.

4. DISCUSSION

A series of careful experiments has been carriedout to assess the e�ect of the settling movement ofNH4Cl equiaxed crystals on their growth velocity in

undercooled NH4Cl±H2O melts. The growth beha-viour of free dendrite was possibly observed for twodi�erent undercoolings (1 and 38C) and for two

di�erent melt compositions (30 and 34% ofNH4Cl). The experimental set-up allowed eachselected crystal to be followed over a relatively long

distance during a large period of time and to recordcontinuously its position and size during the fallthanks to a moving video camera device. The mainexperimental results concern three types of e�ect:

(a) the e�ect of undercooling on the average den-drite growth velocity (Fig. 5); (b) the e�ect of theaverage settling velocity on the average growth vel-

ocity (Fig. 5); (c) the e�ect of the instantaneoussettling velocity on the instantaneous dendritegrowth velocity (Fig. 3). Incidentally, it is note-

worthy that the present results corresponding to anundercooling of about 18C exhibit smaller growthvelocities than those published by Ramani and

Beckermann [9], although the experimental methodis based on the same principles [Fig. 7(a) and (d)].Despite the use of the same alloy with similar com-positions and experimental conditions, the discre-

pancy between both studies is obvious as shown inFig. 7 where the experimental data ranges havebeen reported. The explanation for this discrepancy

might be found in terms of experimental inaccuracyon the imposed undercoolings, especially for lowundercoolings.

We have analysed the present experimentalresults with the theory developed by Ananth andGill [3] valid for an isolated dendrite tip growingagainst an axisymmetric melt ¯ow. The tip growth

velocity was supposed to be the measured averagedendrite growth velocity and the intensity of theaxisymmetric ¯ow to be the settling velocity. The

so-called Stokes ¯ow approximation for the descrip-tion of the ¯uid ¯ow was determined to be the mostappropriate one in the present study. Moreover, it

was decided to ®t the predicted values of thegrowth velocity for given undercoolings and ¯uid¯ow intensities to the experimental ones by modify-

ing the value of the stability constant s*. In therange of the experimental settling velocities [0.007±0.011 m/s], the best ®tting value was found to bes* � 0:081. It is 3.12 times greater than the value

of s* measured by Liu et al. for the purely di�usive

case [10]. With an illustrative aim, the ratio of the

present stability constant s* to the di�usive one s*0

is compared in Table 4 with another estimation

from the experimental work of Lee et al. [5] on

pure succinonitrile. The present results show a ratio

s*=s*0 greater than 1. This seems consistent with

the experimental observations of Lee et al. [1], as

far as the model's assumptions and the di�erences

in materials and in explored ranges allow such a

comparison. However, it must be emphasized that

this empirical result is still unclear if compared to

the extension of the microscopic solvability theory

in the presence of convection [4] which predicts a

decrease of s* with an increasing melt velocity (a

ratio s*=s*0 lower than 1).

As observed, the calculations predict an increase

of the growth velocity with an increasing undercool-

ing and an increasing settling velocity (Fig. 7). In

Fig. 7(b) the theoretical curve calculated for the

measured undercooling of 38C in the case of the

34% NH4Cl±H2O alloy is not too far from the

mean experimental data and shows a similar trend.

In Fig. 7(c), in the case of the 30% NH4Cl±H2O

alloy at 38C of undercooling, the experimental scat-

ter does not allow any appreciation of the coinci-

dence between the experiment and the calculation.

In Fig. 8, for one crystal in each growth condition,

the calculations have been confronted with the time

evolutions of the measured instantaneous sizes

which provide more information than the mean

values. And, in all the cases, the agreement seems

good.

The discrepancies between predictions and exper-

iments could be attributed to two experimental

sources of uncertainty: the undercooling of the

melt, and the size of the crystal. In order to account

for the possible errors related to the ®rst factor, the

theoretical growth velocities have been plotted in

Fig. 7 for some upper and lower limiting values of

the undercooling which de®ne the uncertainty inter-

val. Hence, as it appears in Fig. 7(c), the di�erence

between the two sets of average growth velocities

for the 30% NH4Cl±H2O alloy at 38C of under-

cooling could be explained by a di�erence in the

actual undercoolings. Moreover, concerning the sec-

ond reason of discrepancy, the theoretical sizes

divided by Z2 have been plotted in Fig. 8 in order

Table 4. Ranges of melt velocity normalized to the growth velocityU1/vt and the corresponding ratios of the stability constant in thepresence of convection to the stability constant for the purely dif-

fusive growth regime s*=s*0

Lee et al. [1] Present studySCN NH4Cl±H2O

U1/vt [0±255] [330±1550]

s*=s*0 [1±1.66]$ 3.12%

$From experiments on free dendrite tips.%From experiments on settling equiaxed crystals.

APPOLAIRE et al.: FREE GROWTH OF EQUIAXED CRYSTALS 5861

to estimate the interval in which the size of a spin-ning crystal evolves. Even the greater instantaneous

variations of size with time (and so with settling vel-ocity), which cannot be attributed to the opticalmeasurement uncertainty, are included in this inter-

val. Consequently, it is thought that the ®t betweenthe calculated and the measured values is satisfac-tory.

It seems that the major subject of discussionremains the possibility of using the theoretical workof Ananth and Gill to predict the e�ect of the

settling movement on the growth of equiaxed crys-tals. Hence, it is not certain whether the assump-tions underlying Ananth and Gills' theory are stillrelevant in the present study. First, the melt con-

tained in the tube is surely not an in®nite medium.But, as long as the distance between the dendritetips of the crystals and the tube walls is great com-

pared to the characteristic length of solute trans-port, the melt can be considered as in®nite.Secondly, the assumption of an axisymmetric ¯ow

in the opposite direction of the tip growth appearsvery crude when applied to the lateral tips of anequiaxed crystal, as in Fig. 2(b). Thirdly, in the case

of an equiaxed crystal, it is hazardous to considerthe primary dendrite tips as isolated from the restof the crystal, especially when the melt is ¯owingpast the crystal and when the crystal is spinning.

Fourthly, the ¯uid ¯ow pattern around the tipsdepends on time since the spinning movementinvolves a time evolution of the ¯ow direction rela-

tive to the tips.

5. CONCLUSION

Insofar as the uncertainties linked to the exper-

imental measurements strongly a�ect the compari-son between the experiments and the calculations, itcannot be assured whether the theoretical assump-

tions are too crude. Ananth and Gills' theorytogether with a matched stability constant seem togive a fair prediction even for equiaxed crystalsinvolving a greater complexity than an isolated den-

drite tip. As a result, it could provide a ®rst ap-proximation to predict the growth of equiaxedcrystals in the presence of convection in further

models, taking into account the movement of grainscarried away by all sorts of convective ¯ows.

AcknowledgementsÐThe authors acknowledge Pechineycompany, Creusot Loire Industrie, Ascometal, Aubert &Duval and the French Ministry of Industry for ®nancialsupport of this work. This study has been included in theBrite Euram III Project BE-1112 (European Programmeon Aluminium Casting Technology). Also, the authorswould like to thank Professor Ring, University of Utah,Department of Chemical and Fuels Engineering, for hishelpful suggestions.

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