Interface configurations during the directional growth of Salol—I. Morphology

Acta metall, mater. Vol. 41, No. 2, pp. 409~,24, 1993 Printed in Great Britain. All rights reserved

0956-7151/93 $6.00 + 0.00 Copyright © 1993 Pergamon Press Ltd

INTERFACE CONFIGURATIONS D U R I N G THE DIRECTIONAL GROWTH OF SALOL--I. MORPHOLOGY N. D E Y and J. A. S E K H A R

Department of Materials Science and Engineering, International Center for Micropyretics, University of Cincinnati, Cincinnati, OH 45221-0012, U.S.A. (Received 3 September 1991; in revised form l July 1992)

Abstract--The facet crystallography and morphological transitions in directionally grown transparent salol is studied. The facet cell morphologies are observed in situ in experiments where the velocity of growth is imposed and also during recalascence in the presence of a strong positive temperature gradient. It is found that the growth interface of salol is bounded by various combinations of (I 1l) planes. The choice of a particular ( l l l ) plane is influenced by the root tip undercooling of the interface. At low undercooling values, facet cracking is observed. At a higher undercooling of the interface, the facet tip angle is noted to decrease while retaining the (l I l) planes which bound the interface. A model is presented for explaining the growth over a wide range of interface undercooling and the associated cracking phenomena. Stress assisted faceted dendrites are also noted during growth within glass cells which are separated by less than 12 microns. The origin of the dendrites and the emergence of the related spacings are discussed.

(110) faces of solid salol with its melt at 314 K have been reported as 19 _+ 1, 25 + 1 and 31 _+ 1 degrees, respectively [3]. Because of the very high degree of anisotropy, salol is known to grow from its melt with a distinct macroscopic faceted interface. Other important physical properties of salol are tabulated in Table 1. The value of the dimensionless entropy calculated from the data in Table 1 is 7.2. There is scatter in the growth kinetics of salol as reported by different workers [9-19]. There is also some debate on the nature of the growth mechanism operative under different growth conditions. Most growth rate undercooling relationships have been measured only during free growth of salol crystals. Danilov and Malkin [9, 10] have reported the growth rate of an isolated, well developed polyhedral crystal of salol to be

I. I N T R O D U C T I O N Anisotropy between various crystal planes during growth may arise from differences in the surface energy, solute partitioning coefficient or interface growth kinetics. Such anisotropy may lead to the formation of macroscopic facets. During the free growth of crystals, such anisotropy may cause the crystal to be bound by the slowest growing planes. The interface configuration during directional growth in the presence of a positive temperature gradient is more complex. The role of the anisotropy in microstructure development has been recently examined by C a h n t , In this series of two papers, we examine the interface configuration and defect development during the directional growth of a highly anisotropic material Salol. In Part I, we examine the interface configurations. In Part II, we examine thermal stress cracking and related defects. Salol (benzoic acid 2-hydroxyphenylester) is made up of two benzene rings which are twisted 70 ° relative to one another [1]. It is orthorhombic in structure, with cell dimensions recorded at - 193 K as: a = 23.402 ,~, b --- 11.258/~ and c = 7.961 ~ [1]. Salol has a reported dimensionless entropy, L / ( R T E ) , between 6.2 and 6.7 [2], where L is the latent heat of transformation, R is the gas constant, and TE is the temperature of equilibrium between the solid and liquid. There are appreciable differences in the properties of the solid and liquid phases. Solid salol is not completely wetted by its own melt. Contact angles of (001), (111) and

V = K t exp(-K2/T)exp(-K3/TAT)cm/s

(1)

where K~, K2 and K 3 are constants KI e x p ( - K 2 / T o ) = 5.2 x 10 -2 cm/s

tPresentation at the symposium on the Interface Configuration and Growth of Faceted Solid Liquid Interfaces. TMS symposium, New Orleans, 1991 (org. R. Trivedi and J. A. Sekhar) (no written record except for abstract). 409

Table 1. Physical properties of salol 1261.4 kg/ra3[4l Density (solid) 1250.0 kg/m3 Density (liquid) 214.22 g/mol [4] Molecular weight Melting point 314.68 K [51 1.1 x 10~J/m3[5] Latent heat of solidification 1.18 x 10-Tm2/s[5] Thermal diffusivity(solid at m.p.) 0.91 x 10-7m2/s [5] Thermal diffusivity(liquid at m.p.) 9.12 x 104J/kg [7] Enthalpy 2.889 x l02 J/kg K [7] Entropy 1 x 10-2 Pa-s [7] Viscosity 1.3 x 10-1J/rusK [6] Thermal conductivity(liquid) 3.9x 10 IJ/msK[6] Thermal conductivity(solid) (1212.2 + 9.0706 t/C)(l +.0.01) Specificheat J/kg. C [8] 2.3 x 10-2J/rn2[7] Solid-liquid interfacialenergy 2.5x 10 3j/m2[6]

410

DEY and SEKHAR: THE DIRECTIONAL GROWTH OF SALOL--I

and K3 = 7020(C) 2, AT = undercooling, T = temperature and V = crystallization velocity (cm/s). These kinetics conform to the two-dimensional nucleation law [11]. Pollatschek [12] has reported the growth rate of salol contained in capillaries of 7.5 mm diameter and 1 mm wall-thickness at small undercoolings to be V = 4 × 10 -5 AT 1'7cm/s.

(2)

The equation corresponds closely to screw dislocation type kinetics. Both the above two papers do not report on the specific crystal planes for which the data corresponds. Danilov and Malkin [9, 10] have reported data from the fastest growing edge face, while Pollatschek [12] reported data on a crystal face that had won out in a competition with the slower growing crystals in the direction of the capillary axis. The two papers reported growth velocities differing by 3.5 orders of magnitude at an undercooling of 1.5 K [11]. Neumann and Micus [13] have reported the growth rate of saiol in the capillaries of 0.4 mm diameter and 0.06 mm wall thickness to be V = 1.56 x 10 -5 AT 2"3cm/s.

(3)

Cahn et al, [11] replotted the data of Pollatschek [12] and Neumann and Micus [13] to compare the experimental results with the theoretical expected behavior predicted by Cahn [14] assuming the interface diffuseness concept. It was found that the departure from the AT 2 growth law occurs at undercoolings greater than about 6 K and that a linear growth law is observed at undercoolings in excess of 15 K. It was argued by Cahn et al. [11] that the growth kinetic characteristic of two-dimensional nucleation as well as the observance of a parabolic limiting growth law under conditions in which heat flow is negligible was a strong demonstration of the lateral growth mechanism. The linear growth law at larger undercoolings showed that the growth occurred by a continuous mechanism at an undercooling in excess of 15 K. In the range of interface undercoolings between 6 and 15 K, was believed to be the transitional regime where the growth mechanism slowly changed from lateral to continuous mode. Bouter et al. [5] crystallized salol by unidirectional cooling on a flat metal plate. For small subcooling of the bath, their experimental results showed a kinetic relation for the velocity of growth (V) as V = 2.28 x 10 -4 AT 0"98cm/s

(4)

where AT = T M - Ts where TM is the melting point, Ts is the temperature of the surface of the growing crystal. The crystal face on which the measurement was made was not reported by the authors. Miller [2] studied the growth of salol at interface temperatures of 304, 296, 290 and 286 K. Experiments were conducted in which two microscope cover glasses were sealed together to form a cell of 20 mm square with an interior spacing of 20-25 #m. This cell was filled with salol and placed in a Kofler hot-cold

stage and p r e h e a t e d or cooled to the temperature at which the observation was to be made. In this experiment it was found that transition from faceted to spherulitic growth occurred in the range of undercooling between 25 and 29 K and that the temperature at which maximum growth rate was observed was 295 K. The planes on which these growth rate measurements were made were not reported. Jackson et al. [15] investigated the growth of salol in different undercooling ranges. In the 10 degree undercooling range, salol was seen to grow as large rhombohedral plates in thin walled capillaries similar to those used by Pollatschek [12] and Neumann and Micus [13]. When the experiment was carried out using thin cells (5 x 2.5 x 0.25 em), the salol was seen to grow as a number of different crystals, with large amounts of scatter observed in the growth rates of different facets and different crystals. It was also reported that the growth rate of each facet was frequently seen to increase by a factor of 10 or more when a crack appeared in the crystal. To eliminate the influence of the glass substrate an experimental arrangement was created in which the crystal was made to grow at the bottom of a suspended droplet. With this arrangement, the growth rate observed was lower at the same undercooling value, than that which was measured in capillary tubes or thin cells. Jackson et al. [15] concluded that in the undercooling range of study, the growth rates of salol were strongly dependent on crystal perfection. In the interface undercooling range of about 12 K, a change in growth morphology was reported. At undercooling less than 10 K, the crystal was seen to grow with large facets but at undercoolings greater than 13 K, many more facets were seen to appear, some of which were thought to be of different crystallographic planes. In the undercooling range of 14-40 K, Jackson et al. [15] reported growth rates which were faster than that reported by Neumann and Micus [13]. When growth was carried out in the undercooling range of 50-72 K using a temperature gradient cold stage, they reported that the growth morphology was still faceted, indicating highly anisotropic growth. Once again, the growth planes were not reported. The differences in the measurements by different workers may be due to the influence of heat transfer, solute build-up, anisotropic growth, presence of traces of inert gas, crystal defects and the reporting of measurements from different growth planes. Experiments have been reported by Riveros [16] to determine the rate of crystallization as a function of the orientation and temperature and also to determine whether or not the growth of salol crystal is lateral or continuous over the entire surface. A drop of salol was melted on a microscopic slide and placed on a microscope stage in a temperature regulated chamber. The measured linear rate of crystallization in the direction perpendicular to the (001) face was found to be given by V = (0.03 + 0.01) AT 0"65+0'25 (5)

DEY and SEKHAR: THE DIRECTIONAL GROWTH OF SALOL--I and in the direction perpendicular to (I 10) face it was found to be given by V = (0.10 +__0.03) AT H° +°25

(6)

where, AT is the undercooling in the liquid far from the interface (in K) and V is the linear rate of crystallization (in mm/min). It was also reported that for undercoolings in the liquid larger than 11 K the mechanism of growth at the surface (001) changes from lateral to continuous [9]. Podolinski [17] studied the growth process of salol in varying thicknesses of melt films. The thickness of the melt film was varied from 0.01 to 0.1 mm. In this study it was observed that if the basal plane of the salol crystal was parallel to the film plane, then when the melt supercooling exceeded 1.0 K the crystal acquired the form of a rhomb. The faces did not grow for as long as 3-4 h, as long as the supercooling of the melt did not exceed 2.3-2.4 K. When the basal plane of the growing crystal was perpendicular to the melt film plane, the crystal took the form of platelets and the broadening of lamellar crystals without defects was not detected up to a supercooling of 5 + 0.5 K. However, with defects present on one of the faces, growth was detected at a supercooling of less than 2.3 K. This growth was often seen to stop after a short while. In this study it was also noted that during the growth of crystals from their melt, defects were generated in the face centers when crystals of a certain size were obtained. This size was noted to depend on the melt supercooling. As the melt supercooling was increased, the face size of the crystal decreased as did the defects. It was postulated that formation of defects in the face centers was partially caused by heat removal condition [17]. Recent work has been reported where the growth rates have been reported on specific planes. Jin et al. [18] have observed the surface microtopography of a solid-melt interface during growth using an optical microscope in a thermally regulated stage. They experimentally determined the dependence of growth

velocity (V) of the (102), (012) and (112) faces as a function of the interface undercooling (AT). The threshold undercooling for growth, measured for the (102), (012) and (112) faces were 0.015, 0.55 and 2.04 K respectively. Below this undercooling, the growth rates were found to be less than 10 -6 mm/s. The reported growth kinetics of (102), (012) and (112) faces of salol may be divided into three regimes. At low undercoolings, the faces advance by lateral growth mechanism and at high undercoolings they advance by continuous growth mode. These two regimes are separated by a transitional regime as reported before [11]. Their results from Jin et al. [18] are tabulated in Table 2. Many growth rate measurements have been made on salol by various workers as discussed above, but these have often been carried out in very narrow capillaries for eliminating or reducing heat transport limitations on the growth rate. Morris et al. [19] compared the growth rate of salol growing freely in the melt with the rates of advance of the same crystal facing down a glass capillary tube. It was found that the growth rate in the capillary fell over a period of hours or days until it had reached 1% or less of the free growth rate. This retardation of the growth rate was observed to increase with a decrease in the undercooling of the melt and was seen to vanish for undercooling greater than 5 K. It was also observed that the retardation effect increased with a decrease in the diameter of the capillary tube. An alcoholwater mixture was used to etch the crystal face. The etch-pit density of the free grown face was found to be of the order of 105 cm 2 and in a fully retarded crystal face grown inside a capillary tube the etch-pit density was found to be of the order of 102 cm 2. The etch-pit density was found to increase progressively with the growth rate. It was suggested that the etch-pits were due to dislocations. With this assumption, it was inferred that, if the growth rate of a crystal is a function of the dislocation density, then the retardation in the

Table 2. Growthmechanismand growth kineticsof differentfacesof salol at verylow undercooling[18] Face (102) Undercooling 0.0154).03 0.03-0.10 />0.10 (AT), C V, mm/s 4.1 * 10-4 exp(3.1/TAT) 1.2.10 z. AT Growth Two-dimensional Transitional Continuous mode nucleationgrowth regime growth Face (012) Undercooling 0.55-0.75 0.75-2.04 />2.04 (AT), C V, m m / s 1.3*lO-2exp(210/TAT) I.I*10-3*AT Growth Two-dimensional Transitional Continuous mode nucleationgrowth regime growth Face (112) Undercooling 2.04-2.12 2.12-2.87 />2.87 (AT), C V, mm/s 93.7exp(1457/TAT) 4.3* 10-5*AT Growth Two-dimensional Transitional Continuous mode nucleationgrowth regime growth AM 41/2~G

411

412

DEY and SEKHAR: THE DIRECTIONAL GROWTH OF SALOL--I

capillary was due to the decreasing in the dislocation density. It was also proposed that the freely growing crystals of salol, for some reason were produced in a state of stress because they were often seen to be cracked. When crystals entered a capillary tube then they were assumed to grow in a state of greatly reduced stress and dislocations were no longer introduced and any dislocations already present progressively grew out to the wall, thus decreasing the dislocation density on the growing face [19]. From the above discussion it is clear that in the growth rate measurement in capillaries for an undercooling range of less than 5 K the thickness of the capillary tube is a variable on which the growth rate depends. Ie et al. [20] measured the free growth rate of the slow growing (010) face of salol at an undercooling of less than 5 K. By free growth rate, here, it is implied that the crystal was not grown inside any capillary tube but was grown in a 100 ml large flask. Heat transfer limitations were eliminated by stirring the melt. Their experimental results showed a growth rate of V = 2.85 x 10 -s AT 2"38

(7)

where

ATs=TM-Ts. This is close to the AT 2 dependence which is the function given by the simplest form of the screw dislocation theory. When salol was grown in a 4 mm tube, for the (010) slow growing face they obtained a growth law of the form V -~ k I e x p ( - k 2 / A T ) (8) where AT = TM - T (C). T is the bulk temperature (C) and k I and k2 are kinetic constants. They also reported that in case of prolonged growth in the tube, the growth rate slowly decreases, corresponding to a slow change in kt and k 2. A systematic study of growth morphology and spacing obtained during directional solidification has been carried out by Shangguan and Hunt [21]. Their results indicate a very wide range of stable cell spacings at a fixed velocity and temperature gradient. However the average cell spacing was found to decrease with both temperature gradient and growth rate. Slide thickness effects on the morphology were not explored. Additionally, no crystallographic information on growth was provided. This paper discusses these issues as well as extends the results to higher undercoolings. In all the previous literature the bounding planes during high velocity growth have not been reported. This aspect is discussed in this paper. Cracking type defects observed during the growth of unconstrained salol crystals are also noted during directional solidification. This paper and Part II discuss the thermal origin in of such defects and correlate strain effects with morphological evolution.

2. EXPERIMENTAL

In the present work salol crystals were grown unidirectionally inside a thin glass cell (slides) in the model transport directional solidification apparatus. The apparatus was similar to that described earlier in the studies by Hunt and Jackson [22] and Mason and Eshelman [23]. It has three main functional components: (a) the temperature gradient stage, (b) the motor and drive mechanism and (c) the measuring system. The temperature gradient stage consisted of the hot chamber and cold chamber. These chambers could be moved such that the temperature gradient between them may be adjusted and the solid-liquid interface could be maintained in the gap under a variety of experimental conditions, The temperature of each chamber was fixed by circulating a liquid pumped by an electronically controlled thermal bath. The range of temperatures in the system was 273 K to 523 K for the hot chamber and 223 K to 313 K for the cold one. The thermal stability of each system was within _ 0 . 0 2 K . The sample moved across the thermal gradient by a combined system of a stepper motor and a high precision screw. The stepper motor had 50,000 steps/rev, and was connected to the precision screw by a pulley. The solid-liquid interface morphology and its position was recorded by taking photographs with a camera mounted on a microscope. Both chambers were mounted on a metallic plate which held a movable stage in the zone adjacent to the gap. An optical microscope was mounted on the stage. This set-up allowed a complete view of the gap area between the chambers and movement of the microscope in the measurement plane when desired. A photographic camera was attached to the top of the microscope. For the temperature gradient measurements, a calibrated thermocouple was inserted on one side of the sample and was connected to a chart recorder. A temperature trace as a function of the position in the gap was obtained by traversing the sample with the inserted thermocouple in it, at a constant speed through the gap. A microvoltmeter with +1 #V precision was used to read the thermocouple output. The standard sample cells were made from plain precleaned microscope slides 75 x 25 x 1 mm in dimensions. The two slides were separated by a desired distance of 50, 25 and 12.5 # m by placing two parallel thin strip of teflon of thickness 50, 25 and 12.5 #m respectively between the two slides along the two long edges of the glass slides and leaving enough space between them. The glass plates were then held together with the help of clips and the two long edges were sealed together by inert epoxy. The slides were left for 4 h in a furnace maintained at 333 K for the epoxy to harden. The sample cells were filled with liquid salol using suction. One end of the cell was dipped inside the liquid salol and vacuum suction was applied through the other end. After filling the cell

DEY and SEKHAR: THE DIRECTIONAL GROWTH OF SALOL--I with salol the remaining two small ends were also sealed with epoxy and kept at room-temperature for 8 h for hardening. In a special case, with the intention of making a cell with very small spacing between the glass slides, the two slides were sealed together without any teflon between them. A very small gap formed between the slides because of the entrapped air layer between them. The size of this gap was estimated to be less than 12.5 #m. This cell will be called very thin cell in the ensuing discussions. The purity of salol after the vacuum distillation and filling process was estimated to be greater than 99.5%. Two types of solidification experiment were performed: (1) Unidirectional solidification experiment to examine the steady state growth morphology at different solid-liquid interface growth velocities. In this experiment the temperature of the hot and cold chamber was set to a level such that the solid-liquid interface could be positioned in the gap between the cold and hot chamber. Then, a sample cell with a calibrated thermocouple wire inserted in it was traversed at a constant speed through the gap to obtain a temperature trace as a function of the position in the gap. From this, the temperature gradient in the gap was determined. After calculating the temperature gradient in the gap, the sample cell of thickness 12.5/~m was placed between the hot and cold chamber and was left there for about 4 h without disturbing the set up. Then the cell was traversed at a constant velocity of 5/~m/s from the hot chamber towards the cold chamber with the help of motor and drive mechanism as described above. Initially the solid-liquid interface began moving towards the cold chamber but after moving through some distance, a steady state position within the gap with respect to the cold and hot chamber was noted. This indicated that the solid-liquid interface was also growing with same velocity at which the sample cell was made to move by motor and drive mechanism. Once this steady state was reached, photographs of the solidliquid interface were taken at periodic intervals. Then the same procedure was repeated for the traversed velocity of 20 and 50/~m/s. All the above steps were repeated for the cells of thickness of 25 and 50/~m, and the very thin cell. For the very thin cell additional solidification runs were made with imposed velocities of 1 and 75/~m/s. (2) Unidirectional solidification experiment for recalascent growth: The temperature gradient in the gap was measured as described above. The sample cell of 25/~m was placed between the hot and cold chamber and left there undis.turbed for 4 h, The position of the solid-liquid interface was noted. The cell was moved towards the cold chamber at a speed of 100/z m/s until the interface disappeared into the cold chamber. The movement was then stopped and the interface allowed to recalasce. Photographs of the solid-liquid interface were taken at intervals of 18 s to determine the morphology and the position

413

of the interface at various intervals of time. The same steps were repeated two more times again and photographs were taken at intervals of 3 and 4 s respectively.

3. RESULTS

(a) Steady state growth (i) Morphology and spacing. The typical microstructures observed in the directionally grown salol were functions of the temperature gradient, G, growth velocity, V and the spacing, d, between the glass plates. Figures 1 and 2 show typical patterns at velocity of 5, 20 and 50/~m/s for d = 50 and 25/~m respectively. The temperature gradient in both cases was maintained at 5.4 K/mm. Note that as the velocity is increased the average facet cell spacing reduces. Note also that the apex angle of the facet also decreases with increasing velocity. Figure 3 shows similar microstructure in the very thin cell ( < 12.5/tm nonuniform). In addition to the features noted in Figs 1 and 2, the presence of the secondary arms and the lack of bubbles should be noted. Liquid pockets within the solid are noted in Fig. 3(b). In this cell at a high imposed velocity of 75 #m/s, note also that the apex angle is considerably smaller than that seen in Figs 1 and 2 and that between the facet cells very fine cells may be observed. It has been noted previously [21] that the growth of facet cells in salol is a dynamic process and only a mean spacing may be defined at a given velocity of growth. There is spread along this mean when plotted as a function of time. A similar observation was also recorded by us. Figure 4 is a plot of this mean spacing as a function of the cell separation, d. Each point in the graph represents the mean value of at least 10 photographs as a given setting and therefore represents a large number of cells. Note that the facet spacing, 2, decreases with both velocity and cell separation at a fixed value of temperature gradient. Spacing in the very thin ( < 12.5#m non-uniform) is higher than that of the 12.5/~m separation cell. Table 3 shows the same data plotted in Fig. 4 but additionally gives the highest and lowest values about each mean. Figure 5 shows the mean facet spacing values obtained from this work and from previous publication. The glass slides cell separation, d, in the previous publication [21] was 20/~m [24]. Note that at the lower velocities, 2, may increase with velocity. This is predicted when a screw dislocation mechanism assists growth [25]. Figure 6 is a plot of the average cell tip angle as a function of the interface velocity. Each point in the graph represents the mean value of at least 10 photographs at a given setting of parameters and therefore represents a large number of cells. For the higher separation ceils, i.e. 50 and 25 #m, a linear relationship is noted. As the separation between the glass slides

414

DEY and SEKHAR:

Fig. 1

THE DIRECTIONAL GROWTH OF S A L O L - - I

Fig. 2

DEY and SEKHAR:

T H E D I R E C T I O N A L G R O W T H OF S A L O L - - I

415

,,,i,,,,i,,,,i,,~,1~,,,i,,,,i,,,~1,,, 700.0

,~ ~,s"--"-~ mi'--~-on 12 s

~ 600.0

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.~ 500.0 ~" 400.0 ~ 300.0 >e 2 0 0 . 0 100.0 0.00.0,,,

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,,I,~ 20.0

~,1, ,~,1,,,, 30.0 40.0

I ,,,,I ,,,, I, ,, 50.0 60.0 70.0 80.0

Interface velocity,

}.tm/s

Fig. 4. The measured average cell spacing, 2 as a function of the interface growth velocity for four different cell separations. G = 5.4 K/mm.

1 04

I

I

I

I

r mm I

m

i

I

I

I

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100

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100

Fig. 5. The measured average cell spacing as a function of the growth velocity. Curve A, B and C (Ref. [21]) are for cell thickness 20/am and temperature gradient of 1.1, 1.8 and 3.5 K / m m respectively. Curve D, E, F and G (this report) are for temperature gradient of 5.4 K / m m and cell thickness of less than 12.5, 12.5, 25.0 and 50 # m respectively. Curve H (Ref. [26]) is for temperature gradient of 3.0 K / m m and cell thickness of 150 gm.

Fig. 1. The interface morphology during directional growth of salol, G = 5.4 K/mm, (a) V = 5 #m/s, (b) V = 20 #m/s and (c) V = 50/am/s. The glass slide cell separation was 50 #m. Fig. 2. The interface morphology during directional growth of salol, G = 5.4 K/ram, (a) V = 5 #m/s, (b) V = 2 0 # m / s and (c) V = 50#m/s. The glass slide cell separation was 25/am.

Fig. 3

Fig. 3. The interface morphology in the very thin cell. G=5.4K/mm. (a) V = 5 / a m / s , (b) V = 2 0 # m / s and (c) V = 75 #m/s.

416

DEY and SEKHAR: THE DIRECTIONAL GROWTH OF SALOL--I Table 3. The minimum and maximum 2, noted at a fixed interface growth velocity(V), and glass cell separation (d). The temperature gradient (G) for all cases was 5.4 K/ram V(#m/s) d = 50 #m d = 25/tm d = 12.5~um d < 12.5#m 5

)~min = 6 5 3 / t m 2~,~ = 8 1 6 / t m

2min = 2 5 2 l t m 2 ~ , , = 381 /~m

2~i . = 181 /~m 2~ = 357/~m

)'mi, = 1 6 3 / a m 2~,~ = 4 2 4 # m

20

2 ~ , = 306 # m 2m, ~ = 6 6 3 ~ m

~'min = 153 # m 2m~~ = 2 3 1 / ~ m

2min = 1 7 9 / ~ m 2m,~ = 2 5 0 / t m

~'min = 1 6 0 / ~ m ~,ma~ = 2 4 0 l t m

50

2min = 186 # m 2m=~ = 2 3 3 / ~ m

~'min = 71 I t m 2 , ~ = 170 l t m

~'min = 102 # m 2m,~ = 1 5 8 / ~ m

2mi. = 110 # m 2m~ = 219 ffm

75

--

--

--

3,min = 4 1 / ~ m 3,m,~ = 128 # m

T a b l e 4. T h e m i n i m u m a n d m a x i m u m cell t i p a n g l e , ct, n o t e d a t a f i x e d i n t e r f a c e g r o w t h v e l o c i t y ( V ) , a n d g l a s s cell s e p a r a t i o n ( d ) . T h e t e m p e r a t u r e g r a d i e n t ( G ) f o r all c a s e s w a s 5.4 K / m m V(/~m/s)

d = 50 l t m

d = 25/~m

''''1''''1

d < 12.5/~m

Ctmin = 7 0 . 0 Ctma~ = 113. 0

Ct,,i, = 7 0 . 0 =ma, = 1 1 4 . 0

Ct,,i, = 6 7 . 0 C%ax = 104.5

Ctmi. = 5 7 . 0 ~tm~x = 112.5

20

~%i, = 6 7 . 5 % , x = 111.0

%i, = 66.5 Ctr,a~ = 104.5

%i, = 68.0 ~t~, x = 7 0 . 5

~tmin = 6 6 . 5 ~tm~x = 103.5

50

Gtmin = 6 6 . 0 gm=~ = 6 9 . 0

%i, = 67.0 Ctmax = 6 8 . 0

~tmin = 6 6 . 5 %ax = 6 8 . 0

:tmi. = 6 7 . 0 =ma~ = 102.5

75

--

--

--

d decreases, the average angle diminishes at low velocities, except for the very thin cell. As the velocity is increased, the cell separation d is noted to loose its influence on the average angle. Table 4 shows the highest and lowest angle measured at each point shown in Fig. 6. Some additional features were also noted. In the high separation glass slide cells, cracking along the solid-liquid interface of the macroscopic facet was noted. An example is shown in Fig. 7, V = 5 #m/s, d = 150 # m and G = 3 K / m m , where the clean facet shown in Fig. 7(a) develops a crack during growth as shown in Fig. 7(b). In the very thin cell ( < 12.5 # m non-uniform) half facet cells were noted as shown in Fig. 8(c). Figure 8 (a~l) shows the different morphologies observed for the same velocity 5/~m/s. Note also the secondary facet cells shown in Fig. 8(a). The half facets were seen in this cell up to velocities of 50 #m/s. The tip of these half facets structures always were ahead (lower undercooling) of the complete 110.0

d = 12.5 ~um

5

....

~min = 5 2 . 0 ~t~,~ = 8 7 . 5

faceted dendrites as shown in Fig. 8(c). In Fig. 8(b) a half facet cell is shown which subsequently overtakes the large dendrite. The origin of these half facet cells was often from the break up of a facet dendrite

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70.0

0.0

10.0

20.0 30,0 40.0 Interface velocity,

50.0 l~m/s

60.0

70.0

80.0

Fig. 6. The measured average celt tip angle, ~, as a function of the interface growth volocity for four different cell separations. G = 5.4 K / m m .

Fig. 7. Shows the time sequence of events occuring on a single facet during an imposed growth velocity of 5.0/~m/s and in the presence of a positive temperature gradient of 3 K/mm. The photographs are taken 5 s apart during the growth. (a) Shows a facet with no apparent cracks near the interface. Co) Shows the formation of a sudden crack in a location between the tip and root.

DEY and SEKHAR:

THE DIRECTIONAL GROWTH OF SALOL--I

as may be seen in Fig. 9 where one of the faceted dendrites is beginning to breakup into the half facet cells. The observation of several different morphologies Fig. 8 (a-d) for the same imposed condition is account of (i) differences in the slide separation at

417

the different points and (ii) because of the dynamic nature of microstructure formation i.e. all the different observed morphologies appeared at different times and it was difficult to obtain a clear steady state morphology.

Fig. 8. (a~l) shows the different observed morphologies in the thin cell. V = 5/lm/s and G = 5.4 K/mm.

418

DEY and SEKHAR: THE DIRECTIONAL GROWTH OF SALOL--I The formation of the secondary arms would follow a sequence shown in the schematic of Fig. 10(a--d). The sequence shows that the growth of the dendrite involves a small part of the tip region suddenly growing faster than the rest of the facet. The jog thus formed travels along the original face and leaves behind a secondary arm like formation. (ii) Bubbles were always noted in all experiments except in the very thin cell. The origin of the bubbles seem to be from the liquid and not because of any partitioning process at the solid-liquid interface. All the measurements reported in this section and Part II seem not to be influenced by the bubbles. The bubbles are trapped in the solid, normally in the root region of the facet and may or may not travel with the interface. The spacings and other phenomena noted in the paper are seen to be established irrespective of the presence of bubbles.

(b) Recalascent growth

Fig. 9. The break up of a growing faceted dendrite into a final form which resembled Fig. 8(c).

I"

i"

Fig. 10. Schematicillustration of the formation of secondary arms of the type observed in Fig. 8(a). (a-d) indicate increasing time.

In the recalascent growth experiment the measurement of the interface undercooling and interface velocity was made at the base of the facets. The undercooling is reported, assuming the equilibrium melting point to be 314.6 K. The experiment was carried out by rapidly moving the 25 # m separation slide at 100 pm/s and allowing the interface to disappear into the cold chamber. The movement of the cell was then ceased. Photographs were then taken as the interface re-emerged. Calculation of the undercooling and interface velocity was made as described above. At the very highest undercooling, a large plane front like zone was noted as shown in Fig. ll(a-d). This zone then tended to expand but was quickly engulfed and overtaken by fine facet cells. During regrowth, three facet angles were noted to predominate. These were 31 °, 67.6 ° and 103.8°. The growth of facets belonging to each of these groups was independently recorded. Figure 12(a, b) shows the interface velocities as a function of the undercooling. Figure 12(a) is at the higher value of undercooling and 12(b) is at the lower values. As the undercooling reduces below 7 K, cracks are noted to appear as recalascence continues. Note from Fig. 12(a) that as the undercooling diminishes, a transition from the 31 ° facets to the 67.6 ° facets may occur and finally the 103.8° facets may be noted. This transition to the smaller angles with increasing velocity during directional solidification was also noted in Fig. 6 to indicate a continuous decrease in the average cell tip angle with velocity. Average cell spacings of each of these groups was measured and plotted as a function of the root undercooling as shown in Fig. 13. Although the regrowth of the quenched facet interface corresponded to the morphologies and spacings discussed above, one special case was noted. This was the occurrence of a metastable large spacing facet cell quenched and retained during the rapid growth

DEY and SEKHAR:

THE DIRECTIONAL GROWTH OF SALOL--I

Fig. 11. The observed morphology during the regrowth of an undercooled interface. Regrowth took place under the influence of a positive temperature gradient, G =3.125 K/mm. (a~t) indicate decreasing undercooling.

419

DEY and SEKHAR: THE DIRECTIONAL GROWTH OF SALOL--I

420 (a)

Temperature, 309.68

304.68

K

299.68

294.68

289.68

284.68

80 70

60

•

103.8

•

67.6

41, 31 .o

• •

A

5O o

40

i

30 -?= _E 20 lO 5

10

15 Interface

20 undercooling,

25

30

K

(b) Temperature, 311.18

K

310.18

309.18

308.18

8.0

I=

7.0 6.0 E 5.0

~

4.0

~ ~

3.0

2.0

1.0

,,,l,l,llllllllllrtlll,I

....

I ....

IL,I

0.9

3.5

4

4.5 Interface

5

5.5

underceoling,

6

6.5

K

Fig. 12. Measured relationship between the interface root undercooling and growth velocity during recalascence. (a) At high undercooling values for the three tip angles of the facets. (b) Continuation of (a) at low undercooling values. Tip angle e = 103.8°.

Fig. 14. The facet tip arrangement process during regrowth of a large quenched in 2. A bubble is seen to remain attached to the tip of the facet as it recalasces. (a
process. Figure 14 shows the quenched in large 2 facet cell. During regrowth, the facet cell changed from a 31 ° facet to 67.6 ° facet and fnally to a 103.8 ° facet with a reduction in the interface undercooling.

4. DISCUSSION

Temperature, 309.68 300.0

'

304.68 ' I '

299.68 = ' I '

'

'

K '

294.68 ' I

'

289.68 ' B L ,

'

,

284.68 ,

250.0

200.0

~' 150.0

-=

100.0

50.0 0.0

I 5.0

~ I

~ I , 10.0

I

~ ~ I ~ ~ ' ' I t 15.0 20.0 Interface undercooling, K

I

~ I ' 25.0

'

'

' 30.0

Fig. 13. Measured relationship between the interface root undercooling and primary cell spacing, 2 during recalascence. G = 3.125 K/mm.

(a) Facet tip angle and interface bounding planes The three facet-tip angles mainly observed both during directional growth and during the recalascent experiments were 103.8 °, 67.6* and 31.0 °. The larger angles are noted at low growth velocities (lower interface undercooling) and the smaller angles are noted at the higher growth velocities at the large interface undercooling. Table 5 shows the angles between the family of (111) planes in salol. We note that the angles 103.8 °, 67.6* and 31.0" are formed at the intersection of (111) planes. Thus the interface is always bounded by the (111) planes for undercoolings exceeding approximately 3 K. As the growth conditions change, the interface remains bounded by (111) planes, however by selecting specific combination of planes from the same family it is possible to alter the facet angle. The dynamic process of the change could involve two types of mechanism: (i) the emergence of a new cell nucleated from an undercooled pocket of liquid left behind

DEY and SEKHAR: THE DIRECTIONAL GROWTH OF SALOL--I Table 5. Interplanar angles between different {1 1 1} family o f planes in salol Planes

Angle

(liT) (TTI) (TIT) (liT) (lid (lid

149.0°

(lll)

(lll)

(IIl) (iiT) (TIt) (111) (111) (111)

(ITD (1T D (liT) (Tli) (111) (iil)

(111)

(ill)

(liT) (lll) (TIT) (liT) (TT+) (1T1) (liT)

(TIT) Oil) (Til) (111) (Tll) (111) (lIT)

ated ~ 50 #m and G is in K/mm and V is in mm/s. The above equation was not fitted to data from the very thin glass slide separation cell or the 12.5/~m separated cells. In these cells other effects on account of strain may influence the morphology and is discussed below. For velocities below about 5 ttm/s, the data of Sekhar et al. [26] show an increase in/1 with velocity. This is expected when a screw dislocation mechanism is involved in the growth of the facet faces [25]. An equation predicting an increase in/1 with velocity in this regime has been reported [25] and is given as

112.4°

1o3.8°

/1max ~

76.2°

(ITI) (T11) (liT) (TT1)

2 tan(et/2)[sin(~/2)/x3]l/2Vl/Z/G

(10)

where /1max=maximum cell spacing, x3=kinetic coefficient (Model l i d and ~ = apex angle of angle of facet. This equation indicates a linear relationship between log/l and log V. The data [26] in Fig. 5 shows that such a relationship may be valid.

67.6°

(TTI) (Ill) (TTT) (TIT) (1TT) (TIT) (111) (TTT)

421

31.o+

(c) Cracking

between two growing diverging cells [such an undercooled region of liquid for example is beginning to develop in Fig. 3(b)], and (ii) from a mechanism such as twining or rotation of planes. For the case of the very thin cell the emergence of a different facet-tip angle cell could also occur from the disintegration of a growing dendrite (Fig. 9). During directional growth experiments, the minimum and maximum angles (Table 4) are noted to deviate from the strict angle subtended by the intersection of the (111) planes. It was noted however that predominantly only those angles subtended by the intersection of the (111) planes would be seen. The deviation from these angles is thought to be on account of the photograph being recorded during a rearrangement process or because of strain effects to be discussed below and in Part II.

For large values of 2 (obtained at low interface velocities and undercooling) cracking was often observed in the macroscopic facet. This cracking is related to temperature gradient imposed stresses on the facet and is discussed in detail in Part II. We note here that the results in Part II indicate that the accumulated strain energy per unit volume, Ev, is proportional to (/1 *G) 2, where 2 and G are the cell spacing and average axial temperature gradient respectively. As the spacing increases, the accumulated strain energy on account of the non-linear temperature gradient increases. This strain energy may be often relieved by cracking. The cyclic nature of the undercooling vs velocity diagram in Fig. 12(b) is probably a manifestation of the combined influence of strain energy accumulation and release by cracking. An increase in the strain energy requires a corresponding increase in the interface undercooling. The strain energy may be relieved by cracking, thus causing an increase in the velocity at the same undercooling. Additionally, if the interface is using dislocation for growth, cracking may enhance the dislocation density thus leading to an increase in the growth rate.

(b) Primary spacing 2

(d) Growth

Crystal type = O r t h o r h o m b i c . axis a = 7.961A angle ~t = 90.O~; axis b = I 1.258A angle fl = 90.0°; axis c = 23.402A angle y = 90.0 °.

The data from Fig. 5 may be used in the high velocity regime (i.e. in the regime where/1 decreases with velocity) to fit a single equation relating the average spacing /1avand the variables G, and V. The spacing was noted to decrease with a decrease in the glass cell separation until the glass cell separation was very small i.e. 12.5/~m and below. A best fit equation for 2a~ (mm) for the cells separated by 25 p m and higher was calculated to be /1AV= Ko/(G V)°5

(9)

where/co = 0.06 K l/z s 1/2mm for glass cells separated 20-30 #m, Ko = 0.1 K 1/2 S 1/2mm for glass cells separ-

The undercooling of the interface may be measured at the top (apex) of the facet or the root of the facet. The undercooling comprises of several independently established undercoolings from different effects which predominate at different levels of undercooling. The net undercooling may be related to the velocity of the growth, facet spacing and facet angle by a selection principle which incorporates an optimization process similar to that which is employed for example in eutectic growth [27]. Assuming that the salol is pure, the root undercooling of a facet may be written as

ATroot=ATk + A T ~ . + A T E + A T D + A T v + C

(11)

422

DEY and SEKHAR: THE DIRECTIONAL GROWTH OF SALOL--I

where ATk=average kinetic undercooling of a growing plane, AT~ = undercooling to support all the surface energies, ATE = undercooling to support the strain energy, ATo = undercooling to support the heat diffusive process (this may be considered to be zero because the imposed temperature gradient is adequate to dissipate heat even during the recalascent experiment), A T F = t h e additional undercooling to reach the facet root, and C = a constant which takes into account any undercooling required to support the difference in interface energy between the glass and the crystal and the glass and the liquid. A rigorous solution to this problem will require a strict enunciation of ATr and is left to future work, however the trends in the high and low undercooling regimes are possible to explain with a simple analysis of the various terms in equation (11). In Part II we show that ATE will quickly disappear as the undercooling is increased (2 is decreased). Similarly we may consider ATr to vanish as the undercooling is reduced (2 is increased). Assume C is a constant not influenced by 2 and the ATF is small compared with ATk (this is especially true as large undercoolings) then: For continuous growth ATk = V sin(a/2)/x

(12)

where x = a kinetic constant ATE oc 22

(13)

AT~ oc f ( r / 2 )

04)

where F is the capillarity constant incorporating surface energy. If a linear temperature gradient, G, is assumed then ATE oc G2/tan(~t/2).

(15)

From equation (9) 2 oc 1IV 1/2 for a fixed G and d. A model may now be created for explaining the growth and microstructural features at all undercoolings. At low undercoolings (AT~--~ 0) and ATroot =- V sin(ot/2)/x + CI/V

+ C2G/V 1/2tan(or/2) + C

where C3 is a constant.

(0• Troot) ~ .j

=0

and

\-

~

]z

(18)

Figure 12(a) may be explained as follows. The experiment consisted of imposing a large undercooling and then studying the recalacense. At a very high undercooling diffusion of atoms is slow therefore the velocity is low. This increases as the interface temperature increases. Subsequently the velocity decreases for the 33.0 ° facet cells as the undercooling is relieved. Next the interface reaches an undercooling where the 67.6 ° facet is allowed. When these appear, the interface increases velocity for the same undercooling and then decreases as the interface undercooling decreases as required by the atom attachment kinetic process. A similar transition then occurs to the 103.8 ° angle. It is not entirely clear why the 76.2 ° angle is not an intermediate step. This may have to do with establishing the crystallographic planes required to form this facet angle.

(e) Faceted dendrites (16)

where CI and C2 in equation (16) are constants at a fixed value of G and d and contain the terms to explain the cyclic relationship between V and AT in Fig. 12(b) for a fixed value of ct. Additionally, when cracking takes place the term CI/V--,O, but if the interface is in the transitional regime [11] (i.e. mixed dislocation and continuous) an increase in the velocity may be noted because of the emergence of fresh dislocations to aid growth. At high undercooling (AT~-~ 0 and ATe--, 0) and therefore ATrootwill comprise of the kinetic term and the capillarity term. Substituting for 2 in terms of V we obtain AT~oot= V sin(x/2)x +f(FV1/2/C3)

This implies that the velocity increases as the undercooling is increased as noted in Fig. 12(a). As V increases, AT increases for a fixed angle ~. A transition to lower facet-tip angles is noted with an increase in the undercooling. From equation (17) by reducing the angle ~, the interface may reduce the total kinetic undercooling at the same imposed velocity. A reduction in the angle ~ will reduce the undercooling for a given velocity and so may be preferred. However a reduction in ~ causes a decrease in the spacing 2 and an increase in the interface undercooling (on account of the extra interface energy) for the same volume of solidifying material. This is apparent from the experimental measurements shown in Fig. 13 where for the same undercooling in the high undercooling regime, a large facet angle supports a larger cell spacing, 2. Therefore there is some optimization which may develop. The operating point could be fixed by an optimization principle such as minimum undercooling at the interface. This condition may be given as

(17)

Faceted dendrites are only seen in the very thin separation glass cells. Dendrites are distinguished from facet cells because of the presence of secondary arms. It is believed that these secondary arms form because of the strain effects discussed below and in Part II. The secondary arms are seen only at the low velocities of 5.0 and 20 #m/s. In fact the arms are barely discernable at 20 #m/s. It is also observed as shown in Fig. 3(a) (V = 5mm/s) and (b) (V = 20mm/s) that secondary arm spacing decreases with the increase in velocity of growth. As shown schematically in Fig. 10, the secondary arm spacing [distance between i" and k" in Fig. 10(d)] depends on (a) relative velocity of growth of the plane/j, jk and kl [in Fig. 10(b)] and (b) the angle of the jog and (c) the initial jog dimensions. As mentioned above, secondary arm formation is initiated by a jog

DEY and SEKHAR: THE DIRECTIONAL GROWTH OF SALOL--I suddenly forming at the tip region of the facet, which travels along the original face and leaves behind a secondary like formation. The anglejkl is noted to be 67.6 ° which shows that the kj plane is a (111) plane but different from the 103.8 ° facet tip angle. This gives a remnant secondary arm. This type of jog formation is not seen in the thick cell, instead cracking is seen to happen in the thick cells. As explained in Part II by numerical analysis during the growth of a facet, the tip region of the facet undergoes high shear stress. In the thick cell the material will be more under plain strain condition as compared to that of the thin cell. It is known [28] that the brittleness of materials increases with an increase in the severity of the plain strain condition of triaxiality. Therefore the facets in the thick cell will behave in more brittle manner when compared to that in thin cell. This explains why cracking is not seen in thin cell but is very common in the thick cell. In the thin cell, the material may instead yield under the high shear stress which may lead to the formation of a jog by slip of the planes. Therefore when the glass slide cell separation becomes very small, it appears that a significantly higher level of dislocation creation or mobility may occur instead of cracking. Once a jog is formed, now due to higher level of shear stress closer to the tip as compared to that towards the root (as shown in Part II) the plane kl will have higher level of dislocation density than plane 0" [Fig. 10(b)]. Therefore in the lower velocity growth regime where dislocations assist growth, the velocity of growth of the plane kl will be higher than that of plane ~, in spite of the fact that plane /j is at a lower average undercooling than plane kl, As the velocity of growth increases, the effect of dislocation assisted growth diminishes as compared to the other growth mechanisms and therefore the difference in velocity of growth of the plane O' and kl will decrease which will lead to the decrease in the secondary arm spacing consistent with our observations. This is also the reason for the sudden increase in the primary spacing noted with the decrease in the glass slide cell separation in Fig. 4. When dendrites form in preference to cells an increase in the spacing is noted for solute driven systems [29], primarily because now the tips no longer need to interact as the solute is accommodated between the dendrites. In the case of strain driven secondary arm formation and such dendritic formation, the emergence of the secondary arms cause a mismatch of planes between the dendrites and an increase in the spacing may occur. Liquid pockets may be trapped in the interdendritic space. ( f ) Half facet cells As noted in Fig. 8 half facet cells were also noted in the very thin separated glass slide. In the case of non-faceted cells and dendrites which form because of solute rejection, half cells and dendrites have been noted previously [30, 31] when growth is forced between solute constraining environments or along

423

special surfaces. No such previous record exists for faceted materials. The half cells grew with a higher tip temperature than the dendrites. However the dendrites could overtake the half cells occasionally and vice versa. The tip angle of the half cells was noted to be 56.2 ° which is half of 112.4 ° one of the angles of the (111) plane intersection (Table 5), but normally not encountered during growth. The half cell formation requires further detailed study, however the half cells may be expected to form such that the overall interface undercooling is reduced. When the half cells form, there are no trapped undercooled liquid pockets which for example are seen within the secondary interdendritic spaces. 5. CONCLUSION This work describes morphological rearrangement processes in directionally grown highly anisotropic materials. It is shown that highly anisotropic pure materials like salol grow with a non-planar interface in the presence of a positive temperature gradient. The non-planar interface is bound by various combinations of (111) planes. The interface undercooling determines the specific planes which are allowed. The key result from this work is that the interface bounding planes are influenced by the interface undercooling. The undercooling is mostly comprised of kinetic, surface energy and stress effects. The stress effects which are discussed in detail in Part II may also give rise to cracking and secondary arm development in pure materials. The primary spacing between facet cells is a function of the velocity of growth, temperature gradient and the separation of the glass slide cells. In the high velocity regime salol may be bound by (111) planes which make angles of 67.6 ° and 31 °. At lower velocities the (111) planes intersect at 103.8 °. A model to explain this transition is described. Acknowledgement--The work was carried out as part of a grant AFOSR-90-0308 monitored by Dr Alan Rosenstein.

REFERENCES

I. J. H. Bilgram, U. Durig and M. Wacher, J. Cryst. Growth 57, 1 (1982). 2. C. E. Miller, J. Cryst. Growth 42, 357 (1977). 3. Yu. V. Naidich, V. M. Perevertailo, E. M. Lebovich and L. P. Obushchak, Kristallografiya 18, 377 (1973). 4. Encyclopedia of Chemical Technology, 3rd edn, Vol. 20. Wiley-Interscience. 5. J. A. De Leeuw Den Bouter and P. M. Heertjes, J. Cryst. Growth 5, 19 (1969). 6. A. Vogel and B. Cantor, J. Cryst. Growth 37, 309 (1977). 7. D. J. Kirwan and R. L. Pigford, AIChE Jl, p. 422 (1969). 8. Kurt Neumann and Dara M. A1-Yawir, J. Cryst. Growth I1, 323 (1971). 9. V. I. Danilov and V. I. Malkin, Z. Fiz. Khim. 27, 1837 (1954). 10. V. I. Malkin, Z. Fiz. Khim, 28, 1966 (1954).

424

DEY and SEKHAR:

THE DIRECTIONAL GROWTH OF SALOL--I

11. J. W. Cahn, W. B. Hillig and G. W. Sears, Acta metall. 12, 1421 (1964). 12. H. Pollatschek, Z. Phys. Chem. A 142, 289 (1929). 13. K. Neumann and G. Micus, Z. Phys. Chem. 2, 25 (1954). 14. J. W. Cahn, Acta metall, 8, 554 (1960). 15. K. A. Jackson, D. R. Uhlmann and J. D. Hunt, J. Cryst. Growth 1, 1 (1967). 16. H. G. Riveros, J. Cryst. Growth 2, 173 (1968). 17. V. V. Podolinski, J. Cryst. Growth 46, 511 (1979). 18. Wie-qing Jin, Jing Lin and Hiroshi Komatsu, J. Cryst. Growth 99, 128 (1990). 19. P. J. Morrss, D. Kirtisinghe, R. F. StricklandConstable, J. Cryst. Growth 2, 97 (1968). 20. T. H. Ie and R. F. Strickland-Constable, J. Cryst. Growth 21, 243 (1974). 21. Dongkai Shangguan and J. D. Hunt, Metall. Trans. A 22, 941 (1991). 22. J. D. Hunt and K. A. Jackson, Rev. Sci. Instrum. 37, 805 (1966). 23. J. T. Mason and M. Eshelman, report No. IS-4906, Ames Laboratory, Ames IA (1986). 24. Dongkai Shangguan, private communication on separation of glass slides of the experiments reported in Ref. [25]. 25. D. K. Shangguan and J. D. Hunt, J. Cryst. Growth 96, 856 (1989), 26. J. A. Sekhar, V. Seetharaman and R. K. Trivedi, unpublished research. 27. W. Kurz and D. J. Fisher, Fundamentals of Solidification. Trans Tech Publication (1989). 28. G. E. Dieter, Mechanical Metallurgy, Chap. 7, 3rd edn. McGraw-Hill International edition (1986).

29. R. Trivedi, J. A. Sekhar and V. Seetharaman, Metall. Trans. 20A, 769 (1989). 30. J. A. Sekhar and R. Trivedi, Mater. Sci. Engng A 114, 133 (1989). 31. L. M. Fabietti and J. A. Sekhar, J. Mater. Res. 8, 1987 (1992). APPENDIX Nomenclature G L V R TE T TM Ts AT ATr ATE ATD ATr ATroot ATk 3. 2av 2m~x x x3 F

Temperature gradient Latent heat of transformation Growth velocity Gas constant Equilibrium temperature between solid and liquid Temperature Melting point temperature Temperature of the surface of the growing crystal Undercooling Surface energy undercooling Strain energy undercooling Heat diffusive process undercooling Undercooling to reach facet root Facet root undercooling Kinetic undercooling Apex angle of facet Facet primary spacing Average facet primary spacing Maximum facet primary spacing A linear kinetic constant Kinetic coefficient (Model III) Capillarity constant incorporating the surface energy