Intermediate Phase Nucleation at Diffusion Couple Boundaries
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ABSTRACT The nucleation of intermediate phases at the interphase boundary in diffusion couples is examined theoretically. A variational procedure is described which allows for a self consistent determination of the critical nucleus size and shape, the work of formation of the critical nucleus and the nucleation rate.
INTRODUCTION The formation of an intermediate phase at the boundary of two other dissimilar phases is a phenomenon which is important in many areas of materials processing. Nucleation of intermediate phase precipitates at semiconductor/metal interfaces has important consequences in the metallization of electronic devices and, as a result, nucleation rates and product phases have been determined in a large number of systems [1]. The fabrication of reinforced ceramics by displacement reaction [2] requires the nucleation and growth of the final phases at the original interphase boundaries. In addition, phase formation at the reinforcement/matrix interface plays a critical role in the processing and mechanical properties of metal matrix composites [3]. Early work in the study of intermediate phase formation at interfaces was performed by Walser and Bene' [4] and Bene' [5] who established an empirical rule for predicting the first phase to form out of the many possible phases dictated by the equilibrium diagram. The Bene' rule holds for many systems, but there are notable exceptions. The early work did not address the rate of nucleation and it was not recognized until somewhat later that solute interdiffusion between the two original phases is a necessary prerequisite to the nucleation of a third phase at the boundary. Subsequent discussions of the intermediate phase formation problem were contained in the studies by d'Heurle [6], Thompson [7] and Desre' [8]. In each of these treatments a specific shape for the critical boundary nucleus was assumed. In a recent paper by Hoyt and Brush [9] it was shown that ad-hoc assumptions concerning precipitate shape need not be invoked; the critical nucleus size and shape is uniquely determined by the mathematics of the variational procedure applied to the system free energy change. The purpose of the present note is to extend the work of Hoyt and Brush, hereafter referred to as paper I, to more complicated cases. A full numerical solution to the second order differential equation describing the critical nucleus shape will be employed rather than the perturbation technique developed in I. A numerical solution implies that the shape and work of formation of the critical nucleus can be found at all interdiffusion times and not just in the limit of late times. The next section briefly reviews the necessary mathematics and derives a nucleation rate for an intermediate phase. The results section examines the critical nucleus shape and the nucleation rate as a function of the interdiffusion time. 257 Mat. Res. Soc. Symp. Proc. Vol. 321. @1994 Materials Research Society
THEORETICAL BACKGROUND Consider a diffusion couple whose terminal phases are denoted by ax and 0. Further
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