Ordering Process analyzed by Phase Field Method, CVM and PPM

Abstract

Phase Field Method(hereafter PFM) has been attracting broad attentions as a powerful tool to describe inhomogeneous evolution process in microstructure. The PFM is a continuum model traced back to celebrated Cahn-Hilliard' and Allen-Cahn equations. A key to the success of the PFM is the efficient parameterization of a microstructure through field variable(s) which constitute a free energy functional. Within the PFM, an interfacial boundary is not a specific entity to be separately described, but is merely an inhomogeneous localization of the field variables. The shape of the free energy determines final equilibrium state and the transition path. Therefore, by suitably defining both a free energy and field variable(s), one is capable of describing kinetic evolution of various microstructures, e.g., dendrite growth;' spinodal decomposition," nucleation and growth," crystal grain growth" and the evolution of anti-phase domain structure. ' The latter is the main concern of this article. The field variable in the PFM is a continuum quantity in the sense that atomic information is averaged out over discrete lattice points. Hence, most PFM calculations provide no direct information of the atomic configuration both in the equilibr ium and non-equilibrium states. In reality, however, microstructural evolution process is driven by configurational kinetics through atomic movements, and detailed information fed from an atomistic scale is essent ial for a rigorous description of the time evolution of a microstructure. It is, therefore, desirable to combine PFM with an atomistic theory in a coherent manner. Cluster Variation Method(CVM)8.lo has been recognized as one of the most reliable theoretical tools to derive thermodynamic properties under a given set of atomic interaction energies on a discrete lattice. A key to the CVM is that the wide range of atomic correlations is explicitly taken into account in the free energy functional. A set of cluster probabilities or correlation functions in the free energy functional are employed as variational parameters by which the free energy is minimized to obtain the equilibrium