Eutectic growth: Selection of interlamellar spacings
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I.
INTRODUCTION
THE growth
of eutectic and eutectoid structures has received considerable theoretical and experimental attention since these fine periodic microstructures give rise to improved mechanical properties. [~-7] Furthermore, eutectic or near-eutectic alloys can be directionally solidified to obtain in situ composite structures. Two important parameters of eutectic structures, which can be controlled experimentally, are the relative volume fractions of the two phases and the interlamellar spacing. The volume fractions are controlled to some extent by the composition of the alloy,f8J whereas the eutectic spacing is controlled by the imposed growth rate. Jackson and Hunt, in their already classic paper, tT]developed a detailed theoretical model to relate eutectic spacing with the growth rate for directionally solidified alloys. They solved the diffusion equation by assuming that the lamellar fronts are locally flat and they considered the average curvature of the lamellae only for determining the average capillary undercooling at the solid-liquid interface. Furthermore, they assumed that the diffusion distance ahead of the interface is much larger than the interlamellar spacing so that the periodic diffusion field can be characterized by the solution of the Laplace equation. Under these assumptions, the relationship between the undercooling, AT, at the solid-liquid interface and the eutectic spacing, h, was obtained as A T = Kl,kV + K J , k ,
key aspect, which is not yet resolved satisfactorily, is the principle which dictates the selection of spacing over only a narrow band of spacings. Following the maximum growth rate hypothesis of Zener t41 for undercooled liquids, Tiller trl suggested an equivalent criterion for directionally solidified eutectics. He proposed that the system selects a spacing which gives a minimum undercooling at the interface. This criterion, although used widely in the literature, has no fundamental justification. Jackson and Hunt E71correctly pointed out that of all of the possible spacings predicted by Eq. [1], only a finite range of spacings will be stable with respect to fluctuations in the shape of the interface. We shall denote these theoretical values of the minimum and the maximum stable spacings by hm and hM, respectively, and these are shown in Figure 1. Jackson and Hunt quoted an unpublished work by Cahn which qualitatively shows that h < h m will be inherently unstable to fluctuations in the interface shape. They also showed that as the spacing gets somewhat larger than hm, the larger volume fraction phase develops a depression at the center of its lamellae. Thus, they proposed a limiting maximum spacing, hM, for which the slope of the interface goes to infinity. On the basis of this as-
[1]
where V is the growth rate. Kl and K2 are constant parameters for a given system and are defined in the Appendix. The relationship between the interlamellar spacing and the interface undercooling at a fixed growth rate, as predicted by Eq. [1], is shown schematically in Figure 1.
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