Computer simulation of the effect of coherency strain on cluster growth kinetics
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INTRODUCTION
IN diffusional solid-state transformations, precipitate morphology often changes continuously through dynamic interaction among precipitates and solute atoms. At equilibrium, the morphology of a precipitate can be determined with knowledge of its interfacial energy and coherency strain, both of which are often highly anisotropic, tL2'31 In a dynamic aging process, however, precipitates are rarely at equilibrium. Thus, understanding morphological changes in an aging process requires dynamic simulations as well as a knowledge on the nature of interfacial energy and coherency strain. The purpose of this work is to understand the effect of coherency strain on the precipitate morphology during an aging process. Monte Carlo techniques are employed for dynamic simulations, and to simplify the problem, an isotropic interfacial energy is assumed in the simulations. Accordingly, a dynamic simulation for an aging process requires calculating the coherency strain state associated with arbitrary-shaped precipitates. Eshelby's classic work on elastic inclusions [a] enabled us to understand many strain energy problems associated with solid-state phase transformations. In particular, his equivalency method for ellipsoidal inclusions allowed us to solve the inhomogeneity problem, which has found numerous applications in materials science. [Sj With an ellipsoidal morphology, however, there are two inherent limitations for the purpose of this investigation: (1) the interaction energy between two ellipsoidal inclusions reJONG K. LEE, Professor, is with the Department of Metallurgical and Materials Engineering, Michigan Technological University, Houghton, MI 49931. This paper is based on a presentation made in the symposium "The Role of Ledges in Phase Transformations" presented as part of the 1989 Fall Meeting of TMS-MSD, October 1-5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL. METALLURGICAL TRANSACTIONS A
quires extensive numerical integrations, and (2) two ellipsoids touch each other at only one point, and thus, extension to an arbitrary morphology is prevented. In this work, we show that use of rectangular parallelepipedal inclusions alleviates these two limitations, at least for an elastically homogeneous case. We will derive complete analytical expressions for both the self-strain energy and elastic interaction term associated with parallelepipedal inclusions. It will be further shown that the analytical expressions allow us to calculate the coherency strain energy of an inclusion of an arbitrary shape by partitioning it into regions of parallelepipeds. In the following section, a statistical thermodynamic model is first described. This is followed by a derivation of the coherency strain of a rectangular parallelepiped with an arbitrary misfit strain and the elastic interaction between such precipitates. Monte Carlo results are then discussed for both a two-dimensional (2-D) square lattice and a simple cubic lattice.
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