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I.

INTRODUCTION

IN 1984,

Johnson and Cahn Ill introduced the concept of bifurcation and symmetry arguments to predict how a precipitate shape should change with its size during certain solid-state phase transformations. The equilibrium shape of an isolated precipitate in a two-phase alloy is determined by minimizing the sum of the elastic strain energy and the interfacial free energy. The shape that minimizes the elastic energy is not necessarily equivalent to the one that minimizes the interfacial energy. Because the elastic energy scales with the precipitate volume, V, and the interracial energy scales with the interfacial area, V 2/3, size-induced shape transitions were predicted to occur as a function of the precipitate size provided that the shape that minimizes the elastic energy is different from what minimizes the interfacial energy. The work of the shape bifurcation theory elucidated a number of previous investigations on morphological changes during growth or coarsening of precipitates. [2,3,4] Recent studies on the effect of an applied stress field have revealed more interesting features on the shape bifurcation nature. [5,6l In this work, we extend Johnson-Cahn's theory to those bifurcation problems which occur during a solid-state nucleation. The volume of a critical nucleus increases as the nucleation driving force decreases. [7,8j Thus, depending on the magnitude of driving force, a critical nucleus is expected to exhibit all the bifurcation characteristics if the shape dependency of the elastic energy behaves in a manner opposite to that of the interfacial energy. In order to analytically examine the

J.K. LEE, Professor, is with the Department of Metallurgical and Materials Engineering, Michigan Technological University, Houghton, MI 49931. M.H. YOO, Senior Scientist, is with the Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831. This paper is based on a presentation made in the "G. Marshall Pound Memorial Symposium on the Kinetics of Phase Transformations ~ presented as part of the 1990 fall meeting of TMS, October 8-12, 1990, in Detroit, MI, under the auspices of the ASM/MSD Phase Transformations Committee. METALLURGICAL TRANSACTIONS A

bifurcation nature, we consider two kinds of simple nucleus morphologies, having one shape parameter, under the condition of isotropic interfacial free energy. A formal theory of shape bifurcation during nucleation is presented after a brief review of the classical nucleation theory. Homogeneous nucleation theory is treated first, which is followed by heterogeneous nucleation theory. Because heterogeneous nucleation is so diverse and the associated elasticity problems are difficult, ]Tj the present analysis is limited to a simple model similar to the nucleation of a second-phase particle at a grain boundary. We then show some detailed examples based on the nucleus shapes of cuboidal prisms and oblate spheroids. The elastic strain energy which favors shape transitions is also delineated as a function of a shape variable, the aspect