Estimation of diffusion path slopes at zero-flux plane compositions
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[19]
= f2/3
h = f-i/3
[20]
The dependencies of g and h on f are plotted in Figure 2. Whether or not a system follows shape preserving growth could be tested by quenching alloys at different rates and plotting V q v s (sq) 3/2 to test for a straight line relationship. In the constant growth rate case, g can be approximated when the changes which occur during quenching are small ( e . g . , f > 0.8). Then, the change in surface area is proportional to the mean curvature, K, and the amount of growth, dx, according to 6S
[21]
KSdx
=
which yields g-~
[ l+Sv
\
'
1221
If the measured mean curvature is small with respect to Sv (K/Sv < 0.1), then no correction is necessary because g ~ 1.0. When K/Sv = 2/3,the value for a sphere, Eqs. [22] and [ 19] become identical because spherical growth is both shape preserving and corresponds to constant growth around its perimeter.
A method has been presented for correcting volume and surface area measurements made on dendrites that have had their growth interrupted by quenching. That such corrections are necessary is obvious when considering that discontinuous changes in cooling rate are impossible. Even if the temperature of a crucible surface could be reduced at an infinite rate, there would still be a transient time before the internal sample would achieve a similar rapid cooling ratc. The correction procedure requires the use of Eqs. [11], [19], and [20] to account for the changes in V, S, and S,., respectively. Eq. [ 11] requires knowledge of the phase diagram and the measured dendrite volume, which should be readily available. In addition, the volume of quenched liquid which is inside the dendrite periphery must be measured. An additional estimate is necessary for the composition of liquid vs distance before quenching. Estimates can be made by assuming local equilibrium within the dendrite periphery or by measuring the composition of material just outside the dendrite periphery and assuming the same composition prevails inside the periphery. While it is true that the procedure outlined above is approximate, it may be sufficient to reduce anomalies in experimental data which can be reproduced, but can not be explained by theory. Such corrections are especially important near the front of the mushy zone where the structural changes are most easily modeled yet the corrections, predicted by Eq. [11], are the largest.
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This work was performed at the University of Connecticut in the Institute of Materials Science and was supported by NSF Grant No. DR-8011404.
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REFERENCES
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1. J.D. Verhoeven and E.D. Gibson: Metall. Trans., 1973, vol. 4, pp. 2581-90. 2. P.W. Peterson, T.Z. Kattamis. and A. E Giamei: Metall. Trans. A. 1980, vol. I1A, pp. 1059-65. 3. R . M . Sharp and A. Hellawell: Journal of Crystal Growth, 1970, vol. 6, pp. 253-60.
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Estimation of Diffusion Path Slopes at Zero-Flux Plane Compositions R. VENKATASUBRAMANIAN and M. A. DAYANANDA
g .5
Z
0 0
i .5
I 1 .0
f Fig. 2--Variation of the correction factors for dendr
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