Eutectic spacing selection in the lead-tin eutectic system
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INTRODUCTION
E U T E C T I C alloys are the basis of most casting alloys, which has led to the extensive theoretical and experimental study of them. t~-24] One of the most significant theoretical studies is the Jackson-Hunt (JH) model of the regular, lamellar eutectic structure. 141This model replaces the diffusion field in the liquid with that at a planar front and the two lamellae are considered to have equal average interface undercoolings. This model predicts steady-state growth to occur according to the equation AT = Kjva + K2/~
[1]
Undercooling is a function of both velocity and spacing. Equation [1] is plotted in Figure 1. The diffusion solution and the assumption of growth at the extremum point, hex, allows calculation of the interface shape. 141 Agreement between calculated and observed shapes is good. Thus, the JH model predicts a wide range of spacings for steady-state growth at a constant velocity, v. Several experimental studies of eutectic spacing in the Pb-Sn and other alloys tj~ have shown that the system exhibits only a narrow band of spacings, identified in Figure 1, as the stable range. The principle that controls the selection of spacings over this narrow band has not been established. Early researchers suggested a minimum undercooling criterion. It is possible to observe a distribution of spacings with a unique spacing selection criterion due to dynamical effects associated with achieving the unique spacing when local spacing changes cannot occur by small amounts. However, the adjustment of lamellar eutectic spacing is controlled primarily by the presence and motion of lamellar faults, which induces small changes in spacing. More recent spacing measurements [~u have confirmed that the distribution in spacing at a given velocity is not a consequence of dynamical JINCHENG LIU, Research Assistant, and R. ELLIOTT, Senior Lecturer, are with the Manchester Materials Science Centre, University of Manchester, Manchester MI 7HS, England. Manuscript submitted May 24, 1993. METALLURGICAL AND MATERIALS TRANSACTIONS A
processes and a unique selection criterion. It is now recognized that the fundamental spacing criterion is not unique, and a range of spacings is stable at a constant velocity. Jackson and Hunt TM realized that only a finite range of spacings in Figure 1 would be stable with respect to fluctuations in the shape of the interface. They suggested that when the average spacing is greater than Aext, a fluctuation in spacing would create narrower lamellae in the center of an array. These would grow at a higher temperature and gradually widen reducing the average spacing. If the average spacing is less than the extremum, the narrower lamellae created by the fluctuation would grow at a lower temperature and drop back. Eventually, a lamella would grow out, increasing the average spacing. Thus, the extremum spacing is the smallest spacing that is just stable to a perturbation of the spacing. The maximum spacing was related to the development of a hollow in one of the phases as the lamellar spaci
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