Finding Critical Nucleus in Solid-State Transformations
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NUCLEATION during solid-state reactions (e.g., phase transformations, plastic deformation, fracture, etc.) is by far one of the most difficult problems to deal with in materials research, irrespective of the method of study, i.e., experimental, analytical, or numerical. This is because the actual nucleation process is a rare event and the saddle-point configuration only exists transiently. The existing nucleation models, both classical and nonclassical, assume only a limited number of degrees of freedom for the nucleus and do not provide a complete sampling over all possible structural and compositional configurations. The nonclassical nucleation theory of Cahn and Hilliard[1] based on the gradient thermodynamics of nonuniform systems[2] provides a general framework for treating nucleation. Rather than assuming a particular geometry for a critical nucleus that has bulk properties of the equilibrium product phase and is separated from the parent phase by a sharp interface, it characterizes the nucleus as composition nonuniformity by using a concentration field, and the critical nucleus is determined by the saddle point of the total free energy functional. It was demonstrated[1] that in the limit of
CHEN SHEN, Research Associate, and JU LI and YUNZHI WANG, Professors, are with the Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA. Contact e-mail: [email protected] This article is based on a presentation given in the symposium entitled ‘‘Solid-State Nucleation and Critical Nuclei during First Order Diffusional Phase Transformations,’’ which occurred October 15–19, 2006 during the MS&T meeting in Cincinnati, Ohio under the auspices of the TMS/ASM Phase Transformations Committee. Article published online September 28, 2007 976—VOLUME 39A, MAY 2008
small supersaturation, the nonclassical theory reproduces all features predicted by the classical nucleation theory. The theory can be extended to nucleation of structural nonuniformities as well.[3–6] Since the microstructural features developed during solid-state reactions are often influenced by elastic strain fields that are in general functions of size, shape, spatial orientation, and mutual arrangement of the existing compositional and structural nonuniformities,[7] a rigorous treatment of nucleation in solids requires a selfconsistent description of the interactions between the nucleus and the pre-existing microstructure without any a priori assumptions. Based on gradient thermodynamics[2] and microelasticity theory,[7] the phase field approach is a superset of the Cahn–Hilliard description of chemical inhomogeneities and the Peierls (cohesive zone) description of displacive inhomogeneities[8] and therefore can treat nucleation of various types of extended defects produced by both displacive and diffusional processes (for recent reviews, see References 9–13). However, the difficulty in this approach is to locate the exact saddle point in a configuration space of very high dimensions. Analytical approaches, though undoubtedly use
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