Primary spacing selection in directionally solidified alloys

Acta metall, mater. Vol. 42, No. |, pp. 25-41, 1994 Printed in Great Britain

0956-7151/94 $6.00+ 0.00 Pergamon Press Ltd

P R I M A R Y SPACING SELECTION IN D I R E C T I O N A L L Y SOLIDIFIED ALLOYS S. H. H A N and R. TRIVEDI Ames Laboratory, U.S. Department of Energy and the Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011, U.S.A. (Received 7 October 1992; in revised Jorm 6 June 1993)

Abstract--The stability of primary spacing in cellular and dendritic arrays has been examined through critical directional solidification studies in the succinonitrile-acetone system. The existence of a range of primary spacing has been established for steady-state growth under given experimental conditions, and the stability of primary spacing within this range has been confirmed through velocity change experiments. Different mechanisms that are operative in the unstable regime of primary spacing have been quantitatively examined. Besides the mechanisms of cell or dendrite creation or elimination, a new mechanism for spacing changes was also identified in which the adjustment of primary spacing occurred through the lateral motion of cells or dendrites. A theoretical model, based on the mass balance constraint, is developed to correlate primary spacing with processing conditions, and the results are compared with the experimental data.

INTRODUCTION

where D is the solute diffusion coefficient, V the growth rate, F is the capillarity constant given by the ratio of the interracial energy to the entropy of fusion per unit volume, m is the slope of the liquidus, k is the solute distribution coefficient and C t is the composition in the liquid at the interface (or at the dendrite or cell tip). Note that the parameter do is defined for dendritic growth, and it differs from the one defined for a planar front growth in which Ct is replaced with Co/k. Substitution of the values of lD and do, in equation (1) shows that the microstructural length scale, 2, varies inversely with the square root of the growth rate. In contrast to this simple behavior, experimental studies on primary spacing of cells or dendrites show a different scaling law which has made it difficult to establish precisely the physics which characterize primary spacing. The different scaling law presumably occurs due to the important contribution of the thermal gradient which must be taken into account. Different theoretical relationships have been proposed in the literature to correlate primary dendrite spacing with experimental variables. Some of these relationships have been developed to explain the data in a specific set of experiments. The first significant treatment to characterize primary spacing as a function of growth rate, temperature gradient and alloy composition was developed by Hunt [1]. Using the mass balance condition, and assuming that the region close to the tip can be approximated as a part of a sphere, Hunt derived the following relationship between the primary spacing, ~., and the tip radius, R

The solidification microstructure is generally characterized by suitable length scales, such as cellular or eutectic spacing, primary and secondary dendrite spacing, and dendrite tip radius. These microstructural scales, for directional solidification, can be controlled by the proper selection of alloy composition, the growth rate and the temperature gradient. Each of these processing variables influences the physical processes that govern microstructure formation, F o r the solidification of metals or plastic crystals, the important processes are the thermal diffusion, solute diffusion in the liquid, and the capillarity effect at the interface. These physical processes are associated with characteristic lengths, such as the diffusion length, lD, the capillary length, do, and the thermal length, lT. The length scales of microstructures are then related to processing conditions through these characteristic lengths of the physical processes. Many important length scales of microstructures, such as eutectic spacing, dendrite tip radius, dendrite core diameter, and initial dendrite secondary arm spacing, have been found to be proportional to the geometric mean of the diffusion and capillarity lengths. If 2 represents one of these length scales, then 2 oc ~ (1) where we define the diffusion length, ID, and the capillary length, do, as lD = 2 D / V

(2a)

and do =

r /mC,(1 - k )

(722 R

(2b) 25

-4~

[

taCt(I-k)+-

~

(3)

26

HAN and TRIVEDI: SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION

For dendritic growth, the Hunt model can be simplified since the first term on the right hand side of equation (3) is dominant, and the value of R is given by the dendrite tip stability criterion [2] [( FD ~(C0 ~1/2 4) R = --~VkATo]\Ctj j ~ where tr* is the dendrite tip selection parameter for the steady-state dendrite growth [3] and AT0 is the freezing range of the alloy. Substituting equation (4) into equation (3), gives q/32rDk/I/' G-I/2V-I/4AT~/4 V (5) = L~ j where we have taken Co ~ Ct for dendrite growth at low velocities. Another theoretical relationship between the primary spacing and tip radius was also proposed by Kurz and Fisher [4] who assumed the shape of a cell or a dendrite to be a half ellipsoid of revolution. They obtained the same values of exponents for V, G and AT0 for dendritic regime as those given by equation (5). Theoretically, neither the spherical nor the elliptic front will give steady-state solution, Several experimental studies [5-27] have been carried out in various alloy systems to characterize primary spacing. These results, have been analyzed as 21 ~ V-aG-b, and the values of the exponents a and b obtained from these studies are summarized in Table 1. The data in Table 1 show that the values of exponents vary over a. wide range. Some of these values are of course influenced by the presence of convection in the liquid. However, the exponent on velocity also varies in experiments carried out in transparent systems under negligible convection conditions, which shows that the current theoretical models are not consistent with the experimental results, The major aim of this paper is to report results on critical experiments on primary spacing that can 2

give some insight into the primary spacing selection process. The models of Hunt [1] and Kurz and Fisher [4] relate primary spacing with the dendrite tip radius. Since experimental studies as well as the rigorous theoretical model of dendrite tip radius show a unique selection process, current theoretical models assume the existence of a unique primary spacing under a given set of experimental conditions. Recently, Warren and Langer [28] have carried out a linear stability analysis of an array of dendrite, and they have concluded that no unique primary spacing is stable at fixed growth conditions. Instead, a band of spacing may be stable so that the primary spacing selection may be somewhat analogous to the eutectic spacing selection where the existence of such a band of spacing has been dearly established in the literature [29, 30]. The existence of a range of stable primary spacing has also been established by Lu and Hunt [31] through detailed numerical modeling of cellular and dendritic arrays. The principal objective of the present study is to conduct critical experiments that lead to the understanding of several important aspects of primary spacing selection process. (1) We would like to determine if the primary spacing selection is sharp so that there is a unique spacing value under given experimental conditions. If no unique spacing exists, then we would investigate the stable range of spacing. A distribution in primary spacing has been reported earlier [32] in directionally solidified alloys in which significant convection effects are present. Favier et al. [33] have carried out experimental studies in the A l ~ u system under 1 g and under microgravity environment in space. They have clearly shown that the dendrite distribution becomes regular under microgravity environment, whereas it is random under I g condition. Consequently, if one needs to establish the spacing selection criterion, it is essential to carry out experiments under convection free conditions. For this

Table I. Primarydendriteann spacingas summarizedfrom the literature Dendrite matrix Solute (wt%) b a References 0.4 C, 1Cr, 0.2 Mo 0.4 0.2 23 8 Ni 0.19 10 Fe 0.6-1.15 C, 1.1-1.4Mn 0.56 0.25 12 0.035 C, 0.3 Si 0.26 24 2.2-10.1 Cu 0.44 0.43 11 5.7 Cu 0.36 13 0.15 at.% Mg, 0.33 at.% Si 0.54-0.56 0.28 5 AI 0.63 atYo Mg, 1.39at.% Si 0.1-8.4 Si, 0.1-4.8Ni 1-27 Cu, 5 Ag 0.5 0.5 21 1-5 Sn 2-7 Sb 0.42 16 5-10 Sb 0.45 0.75 22 10-50Sn 0.33 0.45 14 Pb 8 Au 0.44 14 10-40 Sn 0.34).41 0.3-0.41 17 8 Au 0.51-0.52 0.43 18 3 Pd 0.55 0.394),42 18 Cyclohexanol H20 0.38 0.5 15 5.5% tool.% Acetone 0.43-0.53 0,37 19 SCN 1.3 Acetone 0.43-0.53 0.36 20

HAN and TRIVEDI: SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION reason, present experimental studies have been carried out in thin samples where convection effects are negligible. (2) If a band of spacing is stable, then we would like to examine if all spacing within this band are equally probable or if there is a certain spacing that is dominant within this band. (3) Dynamical observations have been made to examine various mechanisms that cause unstable spacing to enter the stable spacing band. These studies have been carried out in the succinonitrile-acetone system in which the stability and instability of spacing can be examined in situ. (4) If a stable band of spacing is present, then a given spacing should remain stable over a small range of velocity. Thus, dynamical changes in primary spacing with small increases in velocity have been carried out to examine the stability of a given primary spacing. (5) Finally, the presence of a distribution in primary spacing significantly influences the conditions for the transition from a cellular to a dendritic structure, and this transition has been examined experimentally, EXPERIMENTAL The purification of materials and the preparation of thin sample with a calibrated thermocouple have been described by Han [34]. Since succinonitrile is susceptible to the moisture in the air, the filling of the cell was carried out in a special loading chamber [34]. Due to acetone's high vapor pressure, some acetone is unavoidably lost during the filling procedure, so that the accurate concentration of acetone must be redetermined after filling the cell. This was achieved by using a ring heater, which was placed outside the cell such that the calibrated thermocouple in the cell was at the center of the ring heater. The ring heater melted a circular zone around a very small island of solid that contained the thermocouple. The equilibrium liquidus temperature was taken to be the one at which the small circular solid cystal neither grew nor melted after a long period of time. The final composition of the alloy was obtained from this liquidus temperature by using the phase diagram, The alloy composition was also checked by determining the critical velocity for the planar interface stability, Directional solidification experiments were carried out for succinonitrile-acetone samples with acetone compositions of 0.055 and 0.13wt%. During this experimental study, velocity was the only variable used to cause changes in the growth conditions. The temperatures of the hot and cold baths, and the solute concentration, were keptconstantthroughoutagiven set of experiments. Small variations in temperature gradients (1.9-2.2 K/ram) were found with the change in velocity, and they were measured experimentally, Very low concentration of acetone was used in the experimental study since the threshold velocity for the planar-cellular transition increases as the concentration decreases,

27

Before each run, the entire cell was first held in the hot chamber for a minimum of two hours to ensure thermal and solutal equilibria. Next, the cell was translated into the thermal gradient field to solidify a small volume fraction of the liquid ( ~ 3 mm of the cell was in the cold chamber). The cell was kept stationary for the time necessary to develop a planar interface. Actual directional solidification runs were then made by imposing a desired velocity. Continuous visual observations were made, and photographs were taken at every 30 or 60 s intervals. EXPERIMENTAL RESULTS Experimental studies will be described in four parts: (A) directional solidification runs were made at different velocities, and at each velocity the system was allowed to reach a steady-state. In these studies, the local primary spacings, i.e. individual spacing measurement between the two neighboring cells or dendrites, at each velocity were measured from the photographs taken under steady-state growth conditions. (B) Directional solidification experiments were carried out at a fixed velocity until a steady-state condition was reached. The velocity was then increased in steps, and the range of velocity over which a given array of dendrites remained stable was determined. (C) The spacing adjustment mechanisms during the transient regime were identified and the regimes of local unstable spacings were characterized. (D) Finally, the local spacing for which a cell began to develop sidebranches at a given velocity was characterized to identify the local cell-dendrite transition condition.

(A) Steady state growth For the measurements of primary spacing under steady-state conditions, it was first necessary to ensure that steady-state conditions were achieved in a given experimental run. When the steady state condition was established, the interface remained at the same position when observed under a microscope. Besides this visual observation, the steady state growth conditions for each run were also confirmed by measuring the tip position as a function of time from the photographs taken during the growth [34]. The variations in primary spacing with velocity were examined quantitatively in the succinonitrile0.055wt% acetone and 0.13wt% acetone systems under a constant temperature gradient condition. The primary spacings between neighboring cells or dendrites in a given array were found to vary significantly. For each velocity, over 50 individual or local primary spacing were measured and the histograms of these results are shown in Fig. 1 for cellular arrays and in Fig. 2 for dendritic arrays. The range of cellular and dendritic spacing as a function of velocity, is shown in Figs 3 and 4 for two different compositions. The bar of spacing at each

28

HAN and TRIVEDI: SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION 8 ~

~

12

6

~

~

9

~

4

~" ~

2

~

8

~

~. ~

4

rT

0 210

0 220

230

240

250

260 270

280

110

134

Spacing (gin)

158

182

Spacing (gin)

Fig. 1. Cellular spacing distribution in SCN-0.13 wt% acetone at (a) V = 0.75 ~rn/s, G = 2.06 K/rnrn, and (b) V= 12.0prn/s, G =2.18 K/rnrn. 20, " ~ ' ~ ; ; . ' ~ ' ~ i ~ ' ~ . ~ ~ " ~ " ~ ' ~

12

/

~

.

o

450

~

iiii 558

,,.

666

8

o

774

882

100

120

Spacing, (~tm)

140

160

180

200

Spacing, (l~m)

Fig. 2. Dendrite spacing distribution in SCN~).055 wt% acetone at (a) V = 2 ttrn/s, G = 1.92 K/ram, and (b) V = 15ttrn/s, G = 2.50K/mrn.

velocity represents the range of measured primary spacing, and the filled circle represents the average value of the primary spacing. Note that for succinonitrile-0.055 wt% acetone composition, both cellular

and dendritic structures were observed in different parts of the sample under the same condition of velocity, whereas a sharp transition was observed for the 0.13 wt% acetone sample. !

200

"""

!

t

•~

t 150

1111) I

i

i

10

15

20

Velocity (~m/s)

t i

I

I

i

10

15

20

Velocity ~m/s)

Fig. 3. Variation in (a) cellular,and (b) dendritic spacing with velocity in SCN-0.055 wt% acetone system. G = 2.I K/rnrn.

i

HAN and TRIVEDI: 1000

....

]

SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION

. . . . . . .

I

"

t

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1000

....

i

. . . . . . .

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X Expt-4 +

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,

mm¢ $T~I

" ~

Expt-5 ,

,

,

n

I

10

Velocity (t.tm)

29

J

20

Velocity (.ttrals)

Fig. 4. Variation in cellular and dendritic spacing with velocity in the SCN-0.13 wt% acetone system. G = 1.90 K/

Fig. 6. Different sets of sequential experiments. In each set the velocity was increased in steps.

mm.

(B) Sequential changes in velocity If the distribution in spacing is the result of the existence of a range of stable spacing, then a given spacing must also be stable for a range of velocities, Thus, in the second set of experiments, the stability of an array was investigated by examining the response of primary spacing to the changes in velocity. The steady state distribution of primary spacing was first established at one velocity and then the velocity was increased in small steps in order to determine the range of velocity over which the spacing distribution remains unaltered. Figure 5 shows the results of an experiment in which a steady-state was established at V = 5 #m/s, and then the velocity was increased in steps to 10 #m/s. As the velocity was increased, the primary spacing was found to increase gradually rather than decrease or remain unchanged. In this experiment

40o

,

®

3O0

~

.~ T o

200

.

s

.

.

.

.

7

® ~

l

z0 Velocity(gin/s) Fig. 5. Spacing variation with sequential increases in velocity, Co = 1 wt% acetone and G = 2.15 K/mm.

only a finite range of velocities could be studied since the length of the sample was finite and the diffusion distances at the low velocities were large. Thus, an another run was made in which the initial velocity was chosen to be the same as the final velocity of the previous run, i.e. 10 #m/s, and the velocity was increased in steps to 16 pm/s. The results in Fig. 5 represent two separate runs. Note that the final velocity of the first run and the initial velocity of the second run were the same, but the primary spacing values were observed to be significantly different depending on the manner in which the system was brought to the steady-state at that velocity. A more extensive study was carried out to examine the range of stable primary spacing. In this case, five different directional solidification runs with increasing velocities were made, and the initial velocity in a given run corresponded to the final velocity of the previous run. The results of these experiments are shown in Fig. 6. Note that, in the first two runs, the primary spacing initially increases slightly with the increase in velocity but once a limit of stability is reached it begins to decrease with velocity. The initial primary spacing in a new run was generally found to be significantly smaller than the primary spacing in the previous run at the same velocity. A third set of experiment was carried out with a longer sample to span a wider range of velocity in one experimental run only so that the stability of spacing with increase in velocity can be properly established. Figure 7 shows the variation in primary dendrite spacing with velocity when the velocity was increased in steps. A slight increase in primary spacing was also found initially in the lower velocity region (5-9 #m/s) and then a drop in primary spacing was observed at higher velocities. A decrease in spacing at 11 #m/s was accompanied by the development of a tertiary branch into a primary dendrite. Further increase in velocity resulted in a sharper decrease in primary spacing at

30

HAN and TRIVEDI: SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION 350

.

.

.

.

.

i .

.~

,

•

~

•

i

5

,

i

i

.

.

,

200

.~ 150

.

i , I

l0

i

20

Velocity (pro/s)

Fig. 7. Spacing changes with velocity in succinonitrile1.0 wt% acetone mixture. All experiments carried out in one solidification run. 19/~m/s which was caused by two simultaneously developed new primary dendrites from tertiary arms. These experimental observations identify the range of velocities over which a given primary spacing is stable,

I

O'c) range

0

(XC)av

" (Xc)ts

100

000 ,

,

,

i

I

i

10

20

Velocity (ttm/s)

Fig. 8. Local stability of cellular spacing as a function of velocity with SCN-0.055 Wt% acetone and G = 2.1 K/mm.

stable and may alter if another mechanism to adjust the spacing becomes operative. The regimes of unstable primary spacing for dendritic structures were also determined. In this case, the dendrite elimination occurred when the local spacing was smaller than the stable spacing. How(C) Spacing selection mechanisms in the unstable ever, when the local spacing was larger than the stable regime spacing, tip-splitting was not observed. Instead, a new In the above sets of experiments, we have identified dendrite was created from one of the tertiary branches. the stable regime of primary spacing during the steady- As in the case of the cellular structure, the unstable state growth of cellular and dendritic interfaces, ranges of spacing were characterized by measuring the We shall now examine the regime of unstable spacing largest spacing, (2d)e~ at which dendrite elimination through the study of primary spacing selection during occurred and the smallest spacing, (2d),, for which a the transient conditions. Specifically, we shall examine new primary dendrite was formed from the tertiary the mechanisms that are responsible for the primary arm. These results are shown in Fig. 9. spacing adjustment. In addition to the elimination and creation of a The major mechanisms by which the adjustment cell or a dendrite, we have observed that the primary in cellular spacing occurs are well-documented in the spacing can also adjust through the relative lateral literature. If primary cellular spacings are smaller than stable spacings, then the local spacing is increased by the process of elimination of a cell. In contrast, if the . . . . , local primary cell spacing is larger than the stable spacing, then a new cell is created by the process T • of tip-splitting. Thus, by observing continuously the 2oo I ~~ dynamical process of primary spacing are adjustment dur... ~ { i~ ing the later part of the transient period it is possible to characterize local spacings that unstable. All the local spacing for which cell elimination occurred were recorded, and the largest local spacing, (2c)el at which ~ cell elimination occurred was identified. Similarly, the 100 I (X~l go smallest local spacing at which tip-splitting occurred, # * (2~)~, was determined. These results are shown in o (~a).v Fig. 8 for cellular arrays. These unstable spacing were " (xd}~ measured on photographs that were taken just before * (ka)et the elimination or the tip-splitting process initiated . . . . . ~ Note that within the range of spacing between these 10 20 two unstable limits, there is no change in local velocity0~m/s) primary spacing by the cell elimination or the cell tip- Fig. 9. Range of stable dendritic spacings with SCNsplitting process. However, these spacing may not be 0.055 wt% acetone and G = 2.1 K/ram.

HAN and TRIVEDI:

SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION

31

(a)

(b) 400

!

350

..,.

....

i i

'

• ~.o

oeq

0 0 ~ 0 0 001 0 0 300

"~

!

250

m 200

.-

150

i .................

[] "-[]

!

[]

i

100

i 60

65

........ 70

75

80

85

90

95

Time (min) Fig. 10. Local primary spacing changes with the development of tertiary arm into a primary dendrite. SCN--1 wt% acetone.

motion of cells and dendrites. This relative sidewise motion of cells or dendrites becomes very significant when some defects, such as gas bubbles, are present. Furthermore, the lateral motion is also significant when a cell or a dendrite is eliminated or created. Figure 10 shows the changes in primary spacing during the formation of a new dendrite• Let 20 be the distance between the two neighboring dendrites between which a new primary dendrite forms, and 2 t be the distance between one of the initial dendrite and the new dendrite. The variations in distances 20 and 2t with time were measured and they are plotted in Fig. 10. It is quite evident that the distance 2 o increases with time through the lateral motion of the two dendrites. This lateral motion was also found to propagate through the dendritic array. Similar observations on lateral

motion of cells or dendrites in an array were also observed when a cell or a dendrite elimination occurred. Note the final 2t spacing is smaller than the initial ;to spacing. In addition to the cell or dendrite creation or elimination, the changes in spacing through lateral translation was also found to occur readily if some defects were present. The most c o m m o n defect that causes changes in spacing was found to be the presence of bubbles in the liquid. The interactions of the growing interface with the bubbles that formed in the interdendritic region also caused the cellular or dendritic interface to reorganize. This reorganization occurs through the change in local spacing near the defect and the subsequent lateral propagation of this change along the entire front of an array. Figure 11 shows a sequence of photographs in which the

32

HAN and TRIVEDI:

SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION

Fig. I 1. A time sequence of photographs which show the interaction between the dendritic interface and a bubble from the bulk. presence of bubbles causes significant alteration in the primary spacing distribution. At the location where the bubble encountered the interface, three dendrites interacted with the bubble, and only two dendrites emerged afterthebubbleinteraction. If thesize of the bubble is small then often no change in the number of dendrites occurs, although the presence of bubble causes a lateral perturbation in an array and a change in spacing.

tinuous changes in dendritic array so that dendrites oscillate laterally rather than grow in a fixed direction when defects are present. Experimental observations also show that the presence of a bubble often causes sufficient perturbation in an array to trigger tip elimination away from the bubble and a subsequent creation of a new dendrite.

In order to examine how a defect influences the propagation of an array, the locations of dendrite tips in an array, and the changes in tip positions as a function of time, were measured. The result for one of these experiments is shown in Fig. 12. A significant change in the configuration of a dendritic array is observed when either the dendrite elimination or creation occurs, or when the array encounters bubbles, The long range effect of defects is evident in Fig. 12, where the local perturbation in an array is propagated laterally. The presence of these defects cause con-

When primary spacing of cells in an array is nonuniform, then it may be possible to form dendrites in a regime where local spacings are above some critical spacing for the sidebranch instability to occur. In order to examine the effect of local spacing on celldendrite transition, experimental studies were carried out to measure local spacing where the sidebranch formation was observed. Figure 13 shows the results for the smallest local primary spacing at which sidebranch development was observed for different growth rate conditions. A least square linear regression gives

(D) Cell-dendrite transition

HAN and TRIVEDI:

SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION

33

Axial qrowth direction of dendrites

l - 30

k

~

~

20

IO

[5~

#'X'~ "6~

/"/~ / ' ~

f'~* ~~1~' ?1"~" /-~" t t']h

Tip elimination Bubble

?_

~

#/h

@

//a

Reference point

T.E. B -

/']h

-~o Time

(min)

Fig. 12. A drawing of actual tip positions of dendrites in an array as a function of time. The presence of defects cause significant lateral motion of dendrite tips. Succinonitrile~0.5 wt% acetone, directionally grown with V = 17 #m/s, and G = 2.30 K/ram.

the following equation for the local cell-dendrite transition condition "~C D =

A V-°584

temperature gradient, alloy composition and system parameters. DISCUSSION

(A) Primary spacing selection criterion Experimental studies show the existence of a stable range of primary spacing at a given velocity during the steady-state growth of cells and dendrites. This stable range was characterized by the distributions in local primary spacing, and its existence was confirmed through the velocity change experiments. The band of spacing can be characterized by examining the maximum (2M), average (2av), and m i n i m u m (2~), spacings as a function of velocity. The ratios of the maximum to the average spacing and the average to the m i n i m u m spacing were examined for S C N 0.055 wt% acetone alloy, and the average values of these ratios over the range of velocities examined

(2M/2~) = 1.156,

(2,,./~.m) = 1.180,

(6)

where A is a constant which includes the effects of

were found as

and for cellular growth

(2M/),,v) = 1.1 59, and (,~,vi,~m) = 1.124

for dendritic growth.

The extent of the band of spacing can be characterized by the ratio, /~M/2rn, which is the ratio of the maximum to the m i n i m u m spacing. The average values of this ratio was found to be 1.36 for cellular growth, and 1.30 for dendritic growth. . . . .

'

~ 200 .~ ~,~

x x× ×

100

'

'

'

'

10 Velocity(I.tm/s)

20

Fig. 13. The smallest local cellular spacing with sidebranch instability as a function of velocity in succinonitrile0.055 wt% acetone. G ~ 2.10 K/ram.

34

HAN and TRIVEDI: SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION J

Nomenclature ]" Specing range •

2oo

• Tea'tiary,-(m~

t

~m*s spacing Den. efim.

~

r~

. 100

O0 °

I:

+

0

Ave Cell

D

0 Cell elim.

stmcing

X Cell-l~n Ii~ls.

i

i

i

i

[

lo

2'o

Velocity (lanl/s)

Fig. 14. Spacing vs velocity in the SCN-0.055 wt% acetone system with G = 2.1 K/mm.

The key aspect of the primary spacing development is to distinguish between the following two possible reasons for the observation of a range of primary spacing under given experimental conditions: (1) the primary spacing selection criterion is unique, but the dynamical effects associated with the spacing adjustment process are sluggish so that the array approaches this unique value only very slowly, and (2) the primary spacing selection criterion is not very sharp and a range of primary spacing is stable under a given experimental condition, The dynamical effect associated with the change in spacing can be very sluggish so that it may not be possible to reach the unique spacing value in the finite time associated in a given experiment. Thus, to characterize these dynamical effects, detailed observations on the local spacing adjustment were carried out. The spacing adjustment mechanisms, such as tipelimination, tip-splitting and a new primary dendrite formation from tertiary arms were examined during the transient period. It was found that, for given experimental conditions, a critical spacing exists such that local spacing below this critical spacing are unstable and cell or dendrite elimination occurs to increase the local spacing. Similarly, another critical spacing was observed such that for any spacing larger than this critical spacing the tip-splitting for cells, or creation of a new dendrite from a tertiary arm, occurred so as to decrease the primary spacing. Within these two critical spacings, neither elimination nor creation of cells or dendrite occurred. All the results of the stability of a cellular and a dendritic array are summarized in Fig. 14. These results show the stable range of spacings as well as the unstable regimes for cells and dendrites with respect to the elimination and creation processes. The processes of cell or dendrite creation and elimination were sufficiently rapid compared to the overall time of a given experimental run, so that dynamical effects are not crucial in the operation of these mechanisms,

Experimental studies very often show attempts by a tertiary arm to become a new primary arm. However, the competition from the neighboring secondary branches often does not allow the tertiary branch to become a primary branch until the dendrite spacing was large enough to allow the tertiary branch to grow out without any interruption from secondary branches. Similarly, the tip-splitting of cells did not always create a new cell, rather an oscillatory structure was often observed with repeated tip-splitting, with only one tip propagating forward, as shown in Fig. 15. Such spacings are unstable, although they do not lead to the change in the number of cells or dendrites. Thus; critical spacings corresponding to cell tip-splitting characterize unstable spacings rather than stable spacings. Furthermore, careful observations of results in Fig. 14 show that the observed stable range of spacings

. . . . . . . . . . .

-. . . . . . . . . . . . . . . . . Fig. 15. An oscillatory instability at the side of a cellular interface that does not lead to the formation of a dendritic structure. Succinonitrile-O.055 wt% acetone, V - 13 #m/s and G = 2.14K/mm.

HAN and TRIVEDI: SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION form a narrower band than that given by the two critical spacings for elimination and creation processes. Specifically, the largest spacing at which cell elimination occurs is often smaller than the smallest observed primary spacing. Thus, the question arises as to whether there is another mechanism by which spacing adjustment can occur on a finer scale. The existence of another mechanism of spacing adjustment was found in this study, and this mechanism is evident in the results shown in Figs 10-12. As observed in these figures, a significant lateral motion of dendrites occurs when a significant perturbation is present in the system. The results in Fig. l0 indicate that as the tertiary arm begins to develop into a primary arm, the spacing between the two initial primary dendrites begins to increase through the lateral motion of dendrites. The spacing between the tertiary arm and the parent primary arm also increased with time. When the tertiary arm became a primary arm, i.e. when the position of the new dendrite tip reached the advancing dendritic front, the distance between the old and the new primary arms was significantly larger than half the original unstable spacing. Furthermore, the time scale for the adjustment of spacing through the lateral motion of dendrites was significantly small so that finer spacing adjustment through the lateral motion does not seem to be a critical barrier for the spacing adjustment process. The creation of a new primary arm from a tertiary branch not only caused the lateral movement of the neighboring dendrites, as shown in Fig. 10, but this disturbance was propagated through the entire array so that lateral motions of all dendrites in an array occurred. Thus the basic notion that a range of spacing exists since the local spacings can only increase by a factor of two or decrease by half due to cell or dendrite elimination or creation is not valid. The lateral translation of an array, observed during the formation of a tertiary arm, was also observed during the cell or dendrite elimination process as well as during growth if the array encountered defects such as particles or air bubbles. Our experimental results show that dynamical effects are not the primary reason for the existence of a distribution in primary spacing. Rather, a range of primary spacings is stable. The existence of a range of stable spacing was confirmed through two sets of experiments which showed (1) the presence of a stable band of spacing at a given velocity, and (2) the presence of a band of velocity over which a given spacing was stable. Both these observations, coupled with the observation of relatively faster spacing adjustment dynamics that occur during the transient period, show that no unique spacing selection criterion operates for primary spacing, and an array with a band of spacing is stable under given experimental conditions. This conclusion is consistent with the recent theoretical analysis of Warren and Langer [28] and the numerical analysis of Lu and Hunt [31].

35

Although a band of spacing is stable under a given set of experiments, some selection process does indeed occur within this band. This is evident in the histograms of spacing, Figs 1 and 2, which show that all spacings within the band are not equally probable, and a peak in spacing distribution is observed. ~ ~j~!~:

.........................

'

~

......

Fig. 16. Rearrangement of dendrites into different spacings after an encounter with the bubble.

36

HAN and TRIVEDI:

SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION

Furthermore, our velocity change experiments, Figs 5 and 6, show that when the velocity is increased from zero to the desired value, then primary spacings which are near the lower limit of stable spacings are observed. On the other hand, if the velocity is increased gradually, primary spacings that are closer to the higher limit of stability tend to form. The precise physics which causes the selection of primary spacing within the stable band needs to be investigated further,

(B) The effects of lateral translation and bubble on the dynamics of spacing adjustment In the dynamical experiments in which the velocity was increased in steps, one would expect the spacing to remain the same or decrease with the increase in velocity. However, spacing increases were observed with increases in velocity, as seen in Fig. 5. Reasons for

this increase in spacings were investigated, and two possible mechanisms were found to be responsible for this increase in spacing with velocity. The first contribution to the increase in dendritic arrays during growth was lateral translation of dendrites caused by the presence of defects. In contrast to what was expected, tip oscillation around [001] direction was observed, such as those seen in Fig. 12, which resulted in a finite increase in spacing with time. This lateral migration was also studied under the steady state condition to ensure that this effect is inherent and not just due to the dynamical changes in velocities. A small but finite deviation from the growth direction was also observed even during the steady state growth when bubbles were present in the system. It should be noted that in the transient regime this lateral movement was influenced by the presence of defects like a bubble, as well as other spacing adjustment

Fig. 17. A time sequence showing the interaction of a bubble with a dendritic array. The interaction between the bubble and the dendrite causes an elimination of a dendrite that is slightly away from the bubble.

HAN and TRIVEDI: SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION mechanisms such as elimination or creation of cells or dendrites. Another possible cause for the increase in spacing with velocity increase was due to the bubbles that nucleated in the interdendritic regions. The nucleation of a bubble in the interdendritic space is frequently observed since the lower pressure in this region due to the solidification shrinkage enhances the nucleation process. A bubble, once nucleated, enlarges with time and becomes elongated along the dendrite growth direction and it often undergoes Reighley instability. Small parts of this elongated bubble become and jump towards the interface or into the bulk liquid from the interdendritic region. This small bubble interacts with the growing interface and causes it to reorganize. The interactions of a bubble with the growing dendritic interface can be divided into two types. As shown in Fig. 16, two growing dendrites that were blocked by a bubble merged into one after the encounter resulting in local spacing changes. In the second case, shown in Fig. 17, the bubble merely changed the spacing between the two dendrites which directly interacted with the bubble, although this increase in spacing propagated laterally and caused elimination of one of the dendrites away from the bubble. In all our experimental studies, the presence of bubbles was found to change dendritic spacing, Bubbles that existed in the bulk were much bigger in size than those from the interdendritic regions, and they caused significant changes in dendrite array characteristics. Once the configuration of dendrites in an array was altered by the interaction with bubbles, the reorganized spacing was found not to change with time. From these observations, it is evident that another mechanism of spacing change is through the lateral motion of dendrites due to the presence of a bubble or due to the dendrite elimination or creation process.

~ B x~

/

The theoretical models on primary spacing by Hunt [1]and by K urz and Fisher [4] are approximate in that they assume a simple geometric shape at the tip of the cell or dendrite. The shapes of the tip region, assumed to be spherical or elliptical, do not give shape preserving solutions for the steady state cellular or dendritic growth. Furthermore, the a priori selection of the shape near the tip precludes possible multiple solutions so that they predict a unique value of primary spacing under given growth conditions. In essence, the primary spacing is indirectly selected in these models through the assumed shape. In order to examine the effect of cellular shape on the primary spacing selection process, and to examine the factors that may be important in giving a range of spacings that is observed experimentally, we shall present a simpler model that is based on the global mass balance constraint. The basic cellular spacing model is shown in Fig. 18. For a steady-state growth in a moving coordinate

c

~z

D j

~ [ /

detached L~s°lid

(C) Primary spacing models

37

liquid

Fig. 18. A schematic diagram of the liquid region considered for the mass balance condition. system, the net flux in volume represented by ABCD is zero. From the divergence theorem, the net flux across the boundary ABCD is also zero. Since the flux across the boundaries BC and DA is zero, the flux across AB and CD gives [36]

fR

fc~ Jn dA +

Jn dA = 0

(7)

where dA = 2nr dS for a cell with radial symmetry in which dS is the length of the element of the interface. The normal flux Jn in a moving coordinate system is given by 0C Jn = - D w - - VoC (8)

on Along the interface, AB, Vn = V cos 0 and dS = dr/ cos 0, where 0 is the angle between the cell growth direction and the element on the interface. Also from the flux balance at the interface, -D(OC/c~n) = VnCi(l-k), where C~ is the concentration along the interface. Along the boundary CD, 3, = - VCo, and dS = - d r . Substituting these values in the mass balance equation gives /~2 (~) - J0 kVC~(r)r dr - J~ vCor dr = 0 (9) .,2 Substituting the value of the interface concentration Q (r) = Ct + G [zi(r) 7.t]/m (1 O)

-

and integrating equation (9), we obtain after some rearrangements, the result = i_kCt] [ z ~ ( r ) - z d r dr )~2mC° 1 (11)

f02

8kG

Cod"

The above result can be rewritten in terms of cell tip undercooling, AT, and the equilibrium freezing range, AT0, as f0,/: ~.2AToU1- A ~ 0 ] [zi(r)-z,]r d r 8G k . (12) Let L = ( A T o - A T ) / G , so that L represents the length of the cell from the location of the planar

38

HAN and TRIVEDI: SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION

interface, i.e. from the solidus temperature isotherm, The above equation can then be written as

ff /~

22L

(13)

[zi(r ) --zt]r dr = - - - ~ In order to solve the above equation, the form of the function zi(r ) is required. The precise shape of the interface is indeed complex, and it requires a self-consistent solution of the diffusion problem, In order to gain some insight into the effect of shape on primary spacing we may consider some simple geometrical shapes. If one considers the interface shape to be elliptic, parabolic or sinusoidal, then equation (13) gives the result 2 = ~x/~ (14) where ct is a constant whose value is 12, 16 or 28.1 for elliptic, parabolic or sinusoidal interface, respectively, and R is the tip radius for the corresponding shape, Note that these shapes are arbitrary and considered only to show the general form of primary spacing result. For dendritic growth at low velocities, the dendrite tip radius given by equation (4) can be written in terms of characteristic lengths for solute diffusion and capillarity as

F1Ddol,/2 R = 1_2--~-j .

(15)

Substituting this value of the radius in equation (14) gives a general relationship between the primary spacing and characteristic lengths as 2 = A[lD]l/a[lT]l/2[do] 1/4 (16a) where the constant

Ct V 1 71/4 A = x/g~/[_2--~j/ .

(16b)

IT is the thermal length which is defined as, IT = mCt(l - k)/G ,~ kL. In comparison with the dendritic tip radius or the eutectic spacing model, the primary spacing model not only depends on the characteristic solute diffusion and capillarity lengths, as shown in equation (1), but it must also include another characteristie length, i.e. the thermal length. Experimental results in several organic systems have given the value of a* = 0.02. Substituting this value in equation (16b) gives .4 = 2.236~/~/k. The result of the primary spacing model, given by equations (14) and (16), gives a clear insight into the role of experimental variables on primary spacing, Equation (14) shows that the primary spacing is proportional to the geometric mean of tip radius and cell/dendrite length (or more precisely, the amplitude of dendrite from the unstable planar front location or the solidus isotherm). On the other hand, equation (16) shows how the primary spacing depends upon solutal, thermal and capillary lengths. These characteristic lengths are directly related to experimental variables so that the change in experimental conditions alters appropriate characteristic length and thus

influence primary spacing. Note that the change in velocity alters the solute length only, whereas the change in temperature gradient influences the thermal length only. Both, the thermal and capillarity lengths are influenced by the change in composition which comes through the change in freezing range. The major unknown in the model is the value of the parameter ct which is strongly dependent upon the shape of the interface. The result with different arbitrary shapes indicates that the mass balance constraint can be satisfied by a range of values of the parameter ct, so that it has a unique value only if the shape of the interface is unique. The parameter ct decreases as the cell shape becomes narrower. The theoretical analysis presented in this paper assumes that there is no second phase or eutectic forming in the intercellular or interdendritic region, and that the diffusion in the solid is negligible. In reality, diffusion in the solid must be taken into account when the length of the cell is finite. The influence of diffusion in the solid can be significant for cellular spacing since the depth of the groove is influenced by the magnitude of diffusion in the solid. Seetharaman et al. [37] have shown that small variations in primary spacing can be correlated with the small variations in the groove depth so that primary spacing is correlated with the groove depth and thus to the diffusion in the solid. Furthermore, the above model is only approximate for dendritic growth since the actual dendritic interface behind the tip region is very complex. In this case it is assumed that the complex interface behind the dendrite tip is replaced by a smooth interface with equivalent volume fraction of solid at any given location [1]. The theoretical models of Hunt [1] and Kurz-Fisher [4] for primary dendrite spacing give results that are analogous to that suggested on the basis of characteristic lengths in equation (16). The value of the proportionality constant A in the Kurz-Fisher model is 3.62/x/k, whereas it is 5.32 in the Hunt model. Thus the basic primary dendrite spacing models proposed by Hunt and Kurz-Fisher are consistent with the result obtained from the overall mass balance constraint, although the numerical value of the constant A varies due to the arbitrary assumption of a spherical or elliptical tip shape in these models. Note that neither a spherical nor an elliptic front would give a steadystate growth. We shall now examine equations (13) and (16) in light of our experimental results and discuss the shortcomings of the current theoretical models of primary spacing. Since a range of primary spacing is observed experimentally, a range of stable cell or dendrite shapes is possible which satisfy the mass balance constraint. For dendritic growth, R has a unique value so that the tip composition and temperature are uniquely defined by equation (4). The range of dendrite spacing must then come from the shape of the interface away from the tip which is reflected in the shape parameter ~. It appears that there is a band of steady-state shapes

HAN and TRIVEDI:

SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION

with different c(values that gives rise to stable primary spacings. The importance of the shape factor d! is evident from experimental results under conditions where both cellular and dendritic structures coexist. In this case, the dendrite spacing is always larger than the cell spacing, although the velocity and temperature gradient dependence appears to be the same from the limited experimental data available in the literature for cellular and dendritic growth under convection-free growth conditions. For cellular growth, the basic model is given by equation (14). In this case not only the value of a is different for different shapes, but the value of the cell tip radius may also not be unique. The precise nature of the capillarity term for cellular growth is not yet established, but Karma and Pelce [38] have used an analogy with the Saffman-Taylor fingers to consider the effect of capillarity. Their result can be written in the form given by equation [2, 381 mG, - G = r/(a*R*)

(17)

or (y)(z)-G

=A

(18)

where G, is the concentration gradient at the tip. Karma and Pelce [38] obtained the value of a* = (l/R)*/F, for cellular growth, where Fh is the dimensionless shape parameter which is a function of (i/R). For cellular growth different steady-state cellular shapes may be possible with different F,,, so that the value of a* for cellular growth may not be unique. Equations (14) and (18) contain parameters, CI and a*, and it is possible that both of these parameters may depend upon the steady-state shape of the cellular interface. Detailed numerical models of cellular growth have been developed by Hunt and McCartney [39,40]. These models need to be evaluated for experimental conditions in the succinonitrileacetone system to establish precisely the role of steadystate interface shape, and the effect of steady-shape on the value of CIand a* for cellular growth. (0)

Cell-dendrite

transition

The celldendrite transition has been examined in the literature, both experimentally and theoretically, and approximate models have been developed to interpret the experimental results. The following theoretical criterion has been proposed by Kurz and Fisher [4] for the celldendrite transition Vcd = GDjkATO

(19)

where Vcd is the transitional velocity. Through the analysis of the experimental data in the succinonitrileacetone system, Billia et al. [41] have characterized the cell-dendrite transition by the equation Vcd= ( V,)O.2(v, )“,”

(20)

where V, = (DAT,/kT) and V, = (GD/AT,) are the critical velocities for planar interface stability at high and low growth rates.

39

Whenever cells and dendrites coexist, it is difficult to characterize the cell-dendrite transition since the transition conditions are not unique. We have shown that the cell-dendrite transition is a function not only of the processing conditions, but it also depends upon the local spacing between the cells. For given processing conditions, the cell-dendrite transition would be unique if the spacings were unique. However, since a significant spread in spacing is demonstrated in this study, the celldendrite transition under given experimental conditions should also be a function of local spacing. Experimental results were obtained in this study to characterize critical spacings at which sidebranches form for different velocity conditions and two compositions. Two different results for celldendrite transitions were observed. (1) The results in the SCN0.13 wt% acetone mixture showed a somewhat sharp transition from cells to dendrites with an increase in velocity. The cellular spacing was found to decrease with velocity, and a sudden increase in spacing was observed when cell to dendrite transition occured. as shown in Fig. 4. Such experimental observations have also been made in the succinonitrile-0.95 wt% acetone system by Eshelman et al. [42]. (2) In the SCN-0.055 wt% acetone mixture the transition was characterized by a coexistence zone of cells and dendrites. The cellular spacing was found to first decrease, then increase, and finally decrease with velocity. In this case, in the coexistence zone, the cellular spacings were always slightly smaller than the dendritic spacings. Such results have also been observed earlier in the pivalic acid-O.5 wt% ethanol system [42]. The increase in cellular spacing was attributed by Seetharaman et al. [37] to the transition from low amplitude cells to deep cells. Whether the transition is sharp or diffused can be related to the range of stability of cellular and dendritic spacings. In the lower composition alloy (0.055 wt%), the transition occurred over a range of velocities where both cells and dendrites coexist. As seen in Fig. 14, the difference in the cell and dendrite spacing is relatively small in the velocity region where both cellular and dendritic structures exist. Once the interface forms a cellular array, it finds it difficult to increase the spacing of the entire array, which is required for the formation of dendrites. Thus, in some cases, a cellular structure is retained. However, if a local spacing becomes slightly larger, sidebranch instabilities can form at that location and the disturbance caused by the formation of this sidebranch can propagate laterally giving rise to a dendritic structure within that grain. These dynamic effects indicate the possibility of finite band of stable solutions which may include the solutions for deep cells and dendrites. Experimental results on local spacings at which sidebranches form are given by equation (6). The transition velocity was found to depend not only on the temperature gradient G and equilibrium freezing range, but also on the largest local cellular spacing.

40

HAN and TRIVEDI: SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION

If the primary spacing selection were unique then the cell dendrite transition would also be unique. For higher composition of the alloy (0,13 wt% acetone), the regime of coexistence of cells and dendrite was not observed, and the cell-dendrite transition was found to be sharp. This may be due to the fact that the range of stable spacing is much wider for low composition alloys than that for high composition alloys. A systematic study of the effect of composition on the range of stable cellular and dendritic spacings thus needs to be carried out to quantitatively establish the reason for the sharp and diffuse transitions, For succinonitrile-0.13 wt% acetone alloy, where a sharp transition was observed, the transition velocity was calculated by using the equations developed by Kurz and Fisher [4] and Billia et aL [41]. The calculated transition velocities are 7.37 and 11.35/~m/s, respectively, which are significantly higher than the experimental value which is between 1.0 and 1.2 #m/s. One of the reasons for the diffuse cell-dendrite transition may be due to the preferred growth direction of cells and dendrites. The cell direction close to the cell-dendrite transition is not precisely in the heat flow direction so that cells in different grains will be oriented differently with respect to the heat flow direction. The cells that are oriented farthest from the heat flow direction tend to become unstable and form dendrites [43]. Consequently, some grains would show cells whereas the others will be dendritic. In order to avoid the effect of dendrite orientation on primary spacing, all experimental measurements were made for dendritic arrays that were growing either in the heat flow direction or were within 5° of the heat flow direction. CONCLUSION Several key aspects of primary cell and dendrite spacing have been examined in this paper through critical experimental studies in directionally solidified succinonitrile-acetone mixtures. The major emphasis was placed on the study of the stability of an array of cells and dendrites. This stability was investigated initially by characterizing the observed distribution in primary spacing under steady-state growth condition. Subsequently, stepwise changes in velocity were imposed to examine the stability of an array with respect to the change in velocity. The presence of a band of spacing, rather than a unique spacing, under given set of experimental conditions, was firmly established. Experimental conditions were designed to obtain convection-free conditions so that the distribution of spacing was not due to convection effects but rather due to the fundamental primary spacing selection process. The precise mechanisms that were operative in the unstable regime of primary spacings were investigated through dynamical studies of spacing adjustment in an array, and critical spacings that were stable against these mechanisms were quantitatively determined. In addition to the well

established mechanisms of tip elimination, tip splitting and tertiary arm development, new mechanisms of lateral migration and defect interaction with the growth front were also found to be responsible for the changes in primary spacings. The spacing adjustment processes were found to be sufficiently rapid so that the dynamic effects were not responsible for the existence of spacing distribution. A simple model of primary spacing, based on global mass balance, was developed. This model gave rise to a simple relationship for primary spacing. It was shown that the primary spacing is proportional to the geometric mean between the tip radius and cell or dendrite amplitude. Based upon this model, the basic factors that lead to the nonunique selection of primary spacing have been discussed. Acknowledgements--This work was carried out at Ames

Laboratory which is operated for the U.S. Department of Energy by Iowa State University under contract No. W-7405ENG-82. This work was supported by the Office of Basic Energy Sciences, Division of Materials Sciences. Authors would like to acknowledge valuable discussions with J. S. Langer, J. D. Hunt and W. Kurz. REFERENCES 1. J. D. Hunt, Solidification and Casting o f Metals Book 192. The Metals Society, London, (1979). 2. W. Kurz, B. Giovanola and R. Trivedi, Acta metall (1985). 3. J. S. Langer, Rev. Mod. Phys. (1980). 4. W. Kurz and D. J. Fisher, Acta metall. 29, 11 (1981). 5. D. G. McCartney and J. D. Hunt, Acta metall. 29, 1851 (1981). 6. G. R. Kotler and L. A. Tarshis, J. Crystal Growth. 5, 90 (1969). 7, P. K. Rohatgi and C. M. Adams, Trans. metall. Soc. A.LM.E. 239, 1729 (1967). 8. J. A. E. Bell and W. C. Winegard, J. Inst. Metals 92, 357 (1963). 9. I. Jin, Acta metall. 22, 1033 (1974). 10. P. C. Dann, J. A. Eady and L. M. Hogan, J. Australian Inst. Metals 19, 140 (1974). 11. K. P. Young and D. H. Kirkwood, Metall. Trans. A 6, 197 (1975). 12. H. Jacobi and K. Schwerdtfeger, Metall. Trans. A 7, 811 (1976). 13. M. A. Taha, Metal. Sci. 13, 9 (1979). 14. C. M. Klaren, J. D. Verhoeven and R. Trivedi, Metall. Trans. A 11, 1853 (1980). 15. T. Okamoto, K. Kishitake and I. Bessho, 3". Cryst. Growth. 29, 131 (1975). 16. J. A. Spittle and D. M. Lloyd, Solidification and Casting of Metals, Book 192, p. 15. The Metals Sot., London (1979). 17. J. T. Mason, J. D. Verhoeven and R. Trivedi, J. Cryst. Growth. 59, 516 (1982). 18. J. T. Mason, J. D. Verhoeven and R. Trivedi, Metall. Trans. A 15, 1665 (1984). 19. K. Somboonsuk, J. T. Mason and R. Trivedi, Metall. Trans. A 15, 967 (1984). 20. H. Esaka, Ph,D. thesis, Ecole Polytechnique Federale De Lausanne, Lausanne, Switzerland (1986). 21. T. Okamoto and K. Kishitake, J. Cryst. Growth. 29, 1033 (1975). 22. G. R. Kotler, K. W. Casey and G. S. Cole, Metall. Trans. 3, 723 (1972).

HAN and TRIVEDI:

SPACING SELECTION IN DIRECTIONAL SOLIDIFICATION

23. A. Suzuki and Y. Nagaoka, J. Japan Inst. Metals 33, 658 (1969). 24. M. A. Eshelman, Ph.D. dissertation, Iowa State Univ., Ames, Iowa (1987). 25. J. O. Coulthard and R. Elliott, J. Inst. Metals 95, 21 (1967). 26. M. Ibaraki, T. Okamoto and H. Matsumoto, J. Japan. Inst. Metals 32, 396 (1968). 27. Y. Miyata, T. Suzuki and J. I. Uno, Metall. Trans. A 16, 1799 (1985). 28. J. A. Warren and J. S. Langer, Phys. Rev. A 42, 3518 (1989). 29. V. Seetharaman and R. Trivedi, Metall. Trans. A 19, 2955 (1988). 30. R. Trivedi, J. T. Mason, J. D. Verhoeven and W. Kurz, Metall Trans. A 22, 2523 (1991). 31. Lu and J. D. Hunt, unpublished work, Oxford Univ. (1992). 32. M. A. Chopra and S. N. Tewari, Metall. Trans. 22A, 2467 (1991).

41

33. M. D. Dupouy, D. Kamel and J. J. Favier, Acta metall. 37, 1143 (1989). 34. S. H. Han, M. S. thesis, Iowa State Univ., Ames, Iowa (1992). 35. K. Somboonsuk and R. Trivedi, Acta metall. 33, 1051 (1985). 36. R. Trivedi, Ph.D. thesis, Carnegie-Mellon Univ. (1966). 37. V. Seetharaman, M. A. Eshelman and R. Trivedi, Acta metall. 36, 1175 (1988). 38. A. Karma and P. Pelc6, Phys. Rev. A 39, 4162 (1989). 39. J. D. Hunt and McCartney, Metall. Trans. 15A, 983 (1984). 40. J. D. Hunt and McCartney, Acta rnetall. 35, 89 (1987). 41. B. Billia, H. Jamgotchian and R. Trivedi, J. Cryst Growth 106, 410 (1990). 42. M. A. Eshelman, V. Seetharaman and R. Trivedi, Acta metall. 36, 1165 (1988). 43. R. Trivedi, V. Seetharaman and M. A. Eshelman, Metall Trans. A 22, 585 (1991).