Generalized Nearest-Neighbor Broken-Bond Analysis of Randomly Oriented Coherent Interfaces in Multicomponent Fcc and Bcc
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INTRODUCTION
INTERFACIAL energies play an important role in the nucleation, growth, and coarsening of precipitates and crystal grains. Unfortunately, a sufficient quantity is not available for direct experimental measurement and most of the published data on interfacial energies have been derived from indirect methods. Most prominent is the fitting of the experimental results on nucleation[1–5] or particle coarsening[6–10] to the corresponding kinetic models. The lack of direct experimental data leads to a number of different numerical methods for modeling interfacial energies. In recent years, growing computer capacities opened the door to increasingly sophisticated approaches. Price and Cooper,[11] Rapcewicz et al.,[12] Sluiter et al.,[13,14] Hartford,[15] and Benedek et al.[16] applied first-principle studies, solving the many-body Schro¨dinger equations. Landa et al.[17] and Wynblatt[18] are using the ‘‘glue potential method’’[19] and combining it with Monte Carlo simulations. Staron and Kampmann[20] apply a cluster dynamic approach and Asta et al.[21–23] apply ab initio and cluster expansion methods, in order to calculate interfacial energies. B. SONDEREGGER, Assistant Professor, is with the Institute for Materials Science, Welding, and Forming, Graz University of Technology, A-8010 Graz, Austria, and the Materials Center Leoben Forschung GmbH, A-8700 Leoben, Austria. Contact e-mail: bernhard. [email protected] E. KOZESCHNIK, University Professor, formerly with the Institute for Materials Science, Welding, and Forming, Graz University of Technology, is with the Christian Doppler Laboratory for ‘‘Early Stages of Precipitation,’’ Institute for Materials Science and Technology, Vienna, and the Institute for Materials Science and Technology, Favoritenstrabe 9-11, 1040 Vienna, Austria. Manuscript submitted November 10, 2007. Article published online January 21, 2009 METALLURGICAL AND MATERIALS TRANSACTIONS A
Mishin[24] and others use the embedded atom method (EAM). Whereas these models can predict interfacial energies with good accuracy, they are not practical for implementation with complex simulation algorithms, for example, of microstructure/precipitate evolution, because of computational costs that are too high. Simulation tools, such as MatCalc[25–27] or other Kampmann Wagner–type models,[3] e.g., by Robson et al.,[4,5,28] require fast estimations of interfacial energies for various phases in a given chemical environment. For this task, sufficiently simple and accurate expressions are needed. The classic nearest-neighbor broken-bond (NNBB) concept meets these demands, because the interfacial energy is expressed as a single, closed equation. The only input data (enthalpy of solution) are available from CALPHAD-type thermodynamic databases.[29] Therefore, in the present work, this approach is taken as the conceptual basis for the development of an extended model that takes into account all neighbor interactions as well as general multicomponent solutions. Furthermore, this article does not focus on finding appropr
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