Cluster Variation Method as a Theoretical Tool for the Study of Phase Transformation
- PDF / 2,480,866 Bytes
- 18 Pages / 593.972 x 792 pts Page_size
- 74 Downloads / 195 Views
TION
THE Bragg–Williams approximation (hereafter abbreviated as BW)[1] has been widely employed in phase equilibria calculations, including a number of commercial software packages which employ the BW thermodynamic functions. The wide use of the BW approximation is partly due to a number of advantageous features that it offers, such as its applicability to deal with multicomponent alloys and alloys with low symmetric structures. These features are originating from the mathematical simplicity of a free energy of the BW approximation that is essentially a function of a single parameter, i.e., alloy concentration. In the BW approximation, alloy concentration is equivalent to point probability which is the probability of finding an atomic species at a given lattice point. Considering only the concentration, however, makes it difficult to introduce the information of local atomic configuration. As will be discussed in this paper, atomic interactions between alloy components induce a higher degree of correlation among atomic configuration, which consequently motivates the introduction of cluster probabilities, equivalently correlation functions into the entropy term. Cluster variation method (CVM)[2] was devised by the late Kikuchi as a statistical mechanics method to deal TETSUO MOHRI is with the Center for Computational Materials Science, Institute for Materials Research, Tohoku University, Katahira 2-1-1 Aoba-ku, Sendai 980-8577 Japan. Contact e-mail: [email protected] Manuscript submitted January 14, 2016. Article published online February 13, 2017 METALLURGICAL AND MATERIALS TRANSACTIONS A
with many interacting particles. The entropy term in CVM is written not only as a function of concentration but also of cluster probabilities including pairs and multibody clusters. Computational tasks that are integral to CVM include the free energy minimization and the identification of independent cluster probabilities. Both tasks were not easily implemented in the early years of CVM method since they are computationally expensive. One had to await the development of computers for the CVM application in practical problems. van Baal[3] was the first to demonstrate that CVM can be successfully applied to a phase diagram calculation with higher accuracy of transformation temperatures and reasonable topology of phase boundary which is related to the order of transformation. CVM is formulated in a discrete lattice and is coherent with various electronic structure calculations for the internal energy. As a result, a number of thermodynamic applications of CVM have been established, including first-principles free energy calculations of phase equilibria such as phase diagrams, thermodynamic quantities including heats of formation at finite temperatures, and the lattice expansion with temperature and concentration variations. These applications have been discussed in the previous work, some of which were cited in the author’s comprehensive review article.[4] Several contributions by the author focusing only on Fe-based alloys are also avai
Data Loading...
