Eutectic growth under rapid solidification conditions
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I.
INTRODUCTION
E U T E C T I C growth is controlled essentially by (1) the solute diffusion processes which are necessary to drive the solidification front and (2) capillarity forces, both of which determine the spacing of the precipitating phases. The essential aspects of eutectic growth theory were developed by Zener m and Hillert t2] and published comprehensively in the classic article by Jackson and Hunt (JH). [3~ The JH model was developed under restrictions of low velocities only. Recently, Trivedi, Magnin, and Kurz (TMK) [4] have developed a more general approach which relaxes some of the major simplifying assumptions made by JH so that theoretical predictions can be made in the high growth rate (large Peclet number) regime which is typical, for example, for laser treatment of alloys. The TMK model considered the case in which the velocities were large for the diffusion field but not large enough to alter the boundary condition due to the nonequilibrium effects at the solid-liquid interface. Thus, they considered the case in which the maximum velocity up to which a stable cooperative growth can take place was smaller than the growth rate at which nonequilibrium effects become important. However, many eutectic systems can be grown with sufficiently high velocities where nonequilibrium effects at the interface become important. Thus, the major aim of this article is to develop a more complete theoretical model of rapidly solidified eutectics which takes into account various nonequilibrium effects and, more specifically, the solute trapping at the interface. Furthermore, the TMK model for the diffusion field gave a result in terms of a complex infinite series for the function P that was introduced in a simplified form by JH. r3] In this article we shall also present a simple analytical form of the function P for the range of parameters of practical interest in eutectic growth. It is shown that this simplified version of the complex P function gives a very good overall agreement with the exact series solution so that it is even possible to make
quick estimates with the pocket calculator. Furthermore, this simplified function retains all of the physics of the more exact treatment. The eutectic growth model is developed here for the directional solidification of binary alloys, and it consists of two parts: (1) the solution of the solute diffusion equation in liquid with boundary conditions which include the interfacial energy and the nonequilibrium effect at the interface and (2) the application of a proper selection criterion for the eutectic spacing under given growth rate conditions. The critical ideas that become important at high growth rates will be emphasized in this model. II. FOR
THEORETICAL
LAMELLAR
METALLURGICAL TRANSACTIONS A
GROWTH
Consider the steady-state growth of a eutectic structure in which the diffusion process in the liquid is required for solute partitioning between the two phases. A schematic diagram of lamellar eutectic, which defines relevant length scales and angles needed for th
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