Selection of a length scale in unconstrained dendritic ... - Science Direct

Journal of Crystal Growth North-Holland

,oua+,.lor CRYSTAL GROWTH

132 (1993) 226-230

Selection of a length scale in unconstrained growth with convection in the melt Youn-Woo

Lee,

Ramagopal

Ananth

1 and

William

N. Gill

dendritic

*

Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590, Received

8 March

USA

1993

For the first time the free growth of fully developed succinonitrile, SCN, dendrites is studied experimentally with carefully controlled, well defined, forced convection velocities, V,, in the ultrapure melt up to 1 cm/s, which is about 40 times larger than the velocity due to natural (thermal) convection, V,, and is 300 times larger than the growth velocity of the dendrite, U. Therefore thermal convection and advection have a negligible effect on our experimental data. The selection parameter, (T* = 2ada/uR*, increases by over 50% as the ratio, Urn/u, of the forced convection velocity, U,, to the growth velocity, u, increases from 3 to 300. This result is opposite to the prediction of microscopic solvability theory for a pure material. Our result also is opposite to that reported for binary experiments which support solvability theory and indicate that c * decreases as Urn/u increases up to values of about 19.

Dendritic patterns are formed when an undercooled melt is solidified. At fixed undercooling, a dendritic pattern propagates with a fixed radius of curvature, R, and growth rate, U, for the leading tip. Transport theories, which neglect the Gibbs-Thompson effect, predict a family of solutions in which the product vR is a constant for fixed undercooling, AT, and velocity of the melt, U,. Nature selects an operating state such that the selection parameter, (+ * = 2ad,/uR2, is independent of AT, where (Y is the thermal diffusivity of the melt and d, = T,yC,/L is the capillary length. Here T’ is the melting point, y is interfacial tension, C, is the heat capacity of liquid, and L is the latent heat of fusion. The selection of the operating state has been the subject of numerous studies, as discussed by Langer [ll, and Kessler, Koplik and Levine [2], and is yet to be resolved. These authors considered the Gibbs-Thompson effect, which describes the change in freezing point due to interfacial tension. They predict by ’ ARC0 USA. z Author

Chemical

Co., Newtown

to whom correspondence

0022-0248/93/$06.00

Sq., Pennsylvania should

0 1993 - Elsevier

19073,

be addressed. Science

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means of microscopic solvability theory that anisotropy in the interfacial tension plays a role, through the Gibbs-Thompson condition, in determining u*. On the other hand, Xu [3] indicates that anisotropy is not necessary to determine theoretically the operating state for the propagation of a dendritic pattern. Thus, the exact value of u *, and its dependence on various parameters of the solidification system, does not seem to be settled yet. In our experiments we vary the undercooling, AT, and U, independently and render both advection and thermal convection negligible in the ultrapure melt. This leads to results for U* in ultrapure melt which are opposite to both those predicted by the microscopic solvability theory of Bouissou and Pelce 141 and to the experiments of Bouissou et al. [5] for a binary system which appear to support the theory. Fig. 1 shows a novel, hermetically sealed, apparatus which was designed to create a precisely measured flow field using only a 100 cm3 of ultrapure succinonitrile under vacuum. It consists of a cylindrical cell, which is attached to a moving flange, and a glass capillary, which is attached to a fixed flange. The flanges are separated by a

B.V. All rights reserved

227

Y. W. Lee et al. / Selection of length scale in unconstrained dendritic growth MOTOR

PULLS

AT U,rlcr?I/~c

1

ELDED BALL

FIXED

FREE SCN

BELLOWS

BEARIN CAPILLARY

GROWING DENDRITE

GROWTH

CHAMBER

TRANSLATES VELOCITY

ULTRA

WITH U,

- PURE

SUPERCOOLED MELT

OF SCN

Fig. 1. The forced

convection

growth cell.

bellows, which is welded to maintain a vacuum inside. The cell is set in motion relative to the capillary by a linear actuator which creates a rigid body motion of the melt. The succinonitrile is purified, as effectively and much more rapidly

1o-1

than can be done by zone refining, by vacuum distillation in a packed multistage column with reflux and transferred to the growth cell under vacuum. Then the cell is sealed off from the environment, and the purity of the SCN, is estimated to be better than 99.9999% by measuring its triple point. The cell is placed in a constant temperature bath which is controlled to kO.001 K. Then the crystal is nucleated in the capillary. After the crystal emerges from the capillary into the melt, the cell is set in motion and moves upwards (opposite to the direction of gravity) while the crystal propagates downward. Thus, the flow field generated is different from that in the experiments of Huang and Glicksman [61, where the fluid is driven by gravity and is coupled to AT, the undercooling. The imposed fluid flow in our experiments was varied up to 300 times the growth rate of the crystal and is independent of AT. The patterns formed by the dendrite are observed through a microscope and recorded on a video film. The growth rate and tip radius are measured from the monitor. At least three measurements of v and R are made at each AT and U,. The measurements deviate from the mean values reported here by 2.5 and 5% in v and R respectively.

1 o-1

I I IIll,,

I I IllIll,

VELOCITY OF FLOW (cm/%x)

, I I,,,

NATURAL CONVECTION l

DATA

HUANG AND GLICKSMAN (1981)

*THIS

I -

WORK (1991)

IVANTSOV SOLUTION WITH d = 0.0195

_

NATURAL CONVECTION 10-3 = : FORCED

04 1 o-4 0.01

UNDERCOOLING,

Fig. 2. (a) Effect of forced convection forced convection

A-i- (“C)

I IllIll

/ I I IllIll 0.1

UNDERCOOLING,

, , I I ,,,b 1

10

AT (“C)

on the growth velocity at various undercoolings, AT, of ultrapure succinonitrile. on the tip radius at various undercoolings, AT, of ultrapure succinonitrile.

(b) Effect of

228

K W. Lee et al. / Selection of length scale in unconstrained

Most theoretical studies, the experiments of Huang and Glicksman [6], and our experiments with only natural convection in the melt show that the shape of the tip of the dendrite is very close to a parabola. We find here that the introduction of convective velocities in the melt along the axis of the dendrite, about two orders of magnitude greater than in natural convection, does not alter the shape of the tip and photographs show that the parabolic approximation is still a good one. The leading dendrite was considered to be fully developed after it became isolated and was unaffected by its neighbors, and the results reported here are only for the leading fully developed dendrites. We considered dendrites to be isolated when their tip velocity, radius, sidebranch spacing near the tip and distance from the tip to the first sidebranch were constant and reproducible. Figs. 2a and 2b show the growth rate and tip radius of a steadily growing dendrite as a function of AT. The imposed fluid velocity enhances the growth rate and reduces the radius of curvature of the tip of the dendrite at a fixed undercooling.

dendritic growth

As the imposed fluid velocity is reduced to zero, our data are in excellent agreement with those of Huang and Glicksman [6] and convection is due only to the buoyancy of the melt as discussed by Ananth and Gill [71. When 0.1 5 U, 5 1 cm/s, we estimate that the ratio of the forced to natural convection velocities, Q/U,, ranged from about 3.5 to 40. Using the thermal convection analogy, UN = dm, and the experimental data for R, we estimate UN to be of the order of 0.025 cm/s. The analogy is adequate for order of magnitude estimates, and indeed, it is quite accurate for AT greater than about 0.1 K, as indicated in figs. 3 and 4 of Ananth and Gill [7]. Thus, it appears that we were able to render the flow due to natural convection and advection essentially negligible compared to the forced convection velocity. Here, p is thermal expansion coefficient and g is the acceleration due to gravity. Ananth and Gill [7] and Saville and Beaghton [8] described the effect of forced convection on the growth of the tips of parabolic dendrites into axisymmetric flow of pure melt. They, however, neglected the effect of surface tension. The forced

VELOCITY FLOW

OF

(cmlsec)

0

NATURAL

8

0

PRESENTDATA

A

HUANG

I 1

CONVECTION

DATA

8 GLICKSMAN

I

I

I

I

I

I

2

3

4

5

6

7

8

STEFAN NUMBER ST = AT/(L/C,) x lo* Fig. 3. The stability

constant,

w *, as a function

of Stefan

number

at various

values of the velocity

U, of ultrapure

succinonitrile.

Y W. Lee et al. / Selection of length scale in unconstrained dendritic growth

convection calculations [7] show that the growth Peclet number, P = T/R/a, based on the growth velocity, approaches that predicted by diffusion theory for a given flow Peclet number, Pe = U,R/a, as the Stefan number, St = AT/(L/C,), increases. By extrapolating the data in figs. 2a and 2b, one can estimate the undercooling at which they approach the results predicted by diffusion theory of Ivantsov. For example, with U, = 1 cm/s the effect of convection becomes negligible at AT above 3 K and at U, = 0.1 cm/s the necessary undercooling is at least 1.3 K. This shows that the effect of the moving boundary dominates convection in the melt at large undertoolings for a given U,. The undercooling above which the forced convection effect is negligible increases with increasing U,. Fig. 3 shows clearly that the data for u* = 2ad,/vR2 with forced convection in the melt increase as U, is increased. Also, the increase in u* with U, is more pronounced at smaller AT and at large enough AT the data appear to approach 0.0195 for all U,, which was varied up to 1 cm/s in our experiments. When U, = 0, our data and the data of Huang and Glicksman [6] are essentially the same and give (+* = 0.0195 at large AT. However, they both show a systematic increase in U* as AT is decreased due to natural convection, which becomes increasingly important at lower AT. The data shown in fig. 3 are the mean values and the individual measurements deviate from these mean values by less than 13%. However, the data on u* increase systematically with U, by as much as 50%, except for the two points at St = 0.01 for U, = 0.6 and 1 cm/s which deviate within experimental error. Bouissou and Pelce [4] included the effects of convection, surface tension and crystalline anisotropy and used solvability theory to obtain the length scale and growth velocity. They concluded that l/a * increases with U, and is linear in the term 2

U, d,

--

VW

Pad,,

229

=

2

--

[

fP

1 &l?e

1,114

dog R

V1

Fig. 4. Comparison of ultrapure succinonitrile experimental results for l/a * with theory of Bouissou and Pelce [4].

Reynolds number, U-R/u, v is kinematic viscosity. A direct comparison of the theory with our experiments is given in fig. 4. This figure shows clearly that the theory does not correlate the data very well; l/u * is not linear and the results have a negative rather than positive slope. The negative slope is because u* increases rather than decreases with U, as predicted by solvability theory in ultrapure melt. However at U, = 0 the theory and experiments agree. The present data for u* on a single component ultrapure system exhibit an opposite trend when compared with the data of Bouissou et al. [5] for pivalic acid-ethanol dendrites, which were grown horizontally in a rectangular cell. The binary data support the theory as discussed by Bouissou and Pelce [4], while our single component data do not. This may be due to the presence of ethanol which introduces solutal convection, and to differences in the orientation of the dendrite with respect to gravity.

‘l/l4

i P 3/4 In Re v R 1 where p is the anisotropy, d, is the capillary length, v is the growth velocity and Re is the

This work was supported in part by NYSERDA and NSF Grants CBT8796343 and 8807659.

230

Y. W. Lee et al. / Selection of length scale in unconstrained dendritic growth

References [l] J.S. Langer, in: Chance and Matter, Proc. Les Houches Summer School, Session XLVI, 1986, Eds. J. Souletie, J. Vannimenus and R. Stora (North-Holland, Amsterdam, 1987); J.S. Langer, Rev. Mod. Phys. 52 (1980) 1. [2] D.A. Kessler, J. Kophk and H. Levine, Physico-Chem. Hydrodyn. 6 (1985) 507. [3] J.J. Xu, Phys. Rev. A 43 (1991) 930.

[4] Ph. Bouissou and P. Pelce, Phys. Rev. A 40 (1989) 6673. [5] Ph. Bouissou, B. Perrin and P. Tabeling, Phys. Rev. A 40 (1989) 509. [6] SC. Huang and M.E. Glicksman, Acta Met. 29 (1981) 701,716. [7] R. Ananth and W.N Gill, J. Crystal Growth 91 (1988) 587; 108 (1991) 173; Chem. Eng. Commun. 68 (1988) 1; J. Fluid Mech. 208 (1989) 575. [8] D.A. Saville and P.J. Beaghton, Phys. Rev. A 37 (1988) 3423.