Abstract
First-principles phase equilibria calculations often overestimate an order-disorder transition temperature due to the neglect of local lattice distortion effects originated from the mixture of elements of different atomic sizes. The lattice vibration effects introduced through the Debye-Grüneisen theory within the quasi-harmonic approximation has proven to be quite effective in circumventing the inconveniences. With the preferential enhancement of the stability of a disordered phase by introducing the lattice vibration effects, the transition temperature was reduced considerably. In order to gain further insight into the lattice vibration effects, a systematic investigation of the vibrational free energy of the Debye-Grüneisen theory is attempted on the two-dimensional square lattice which constitutes a prototype study prior to the first-principles calculations on realistic alloy systems. A particular focus of the present study is placed on the effects of Debye temperatures of constituent phases on the transition temperature. It is shown that lattice softening by lattice vibration stabilizes the disordered phase by reducing the energy expended to accommodate atoms of different sizes, which is manifested by the reduction of the curvature of the atomic potentials. It is, however, predicted that an opposite case can also take place. When the Debye temperature of an ordered phase is lower than that of the pure metals, the ordered phase is more stabilized and the inclusion of the lattice vibration effects in the free energy raises the resultant transition temperature.
Similar content being viewed by others
References
T. Mohri, Statistical Thermodynamics and Model Calculations, Alloy Physics, Chap. 10 and references therein, W. Pfeiler, Ed., Wiley-VCH, 2007, p 525-588
R. Kikuchi, A Theory of Cooperative Phenomena, Phys. Rev., 1951, 81, p 998-1003
R. Kikuchi, Space is Continuous-Continuous-Displacement Treatment of Phase-Separating Diagrams, J. Phase Equilib., 1998, 19, p 412-421
R. Kikuchi and A. Beldjenna, Continuous Displacement of Lattice Atoms, Physica A, 1992, 182, p 617-634
R. Kikuchi and K. Masuda-Jindo, Calculation of Alloy Phase Diagrams by Continuous Cluster Variation Method, Comp. Mater. Sci., 1999, 14, p 295-310
H. Uzawa and T. Mohri, Calculation of Short-Range-Order Diffuse Intensity for a Two Dimensional Square Lattice Within Cluster Variation Method, Mater. Trans., 2001, 42, p 422-424
H. Uzawa and T. Mohri, Continuous Displacement Cluster Variation Method in Fourier Space, Mater. Trans., 2002, 43, p 2185-2188
T. Mohri, Theoretical Investigation of Phase Equilibria by the Continuous Displacement Cluster Variation Method, Int. J. Mater. Res., 2009, 100, p 301-307
A. Zunger, First-Principles Statistical Mechanics of Semiconductor Alloys and Intermetallic Compounds, Statics and Dynamics of Alloy Phase Transformations, P.E.A. Turchi and A. Gonis, Ed., Plenum Press, New York, 1994, p 361-419
V. Moruzzi, J.F. Janak, and K. Schwarz, Calculated Thermal Properties of Metals, Phys. Rev. B, 1988, 37, p 790-799
T. Mohri and Y. Chen, First-Principles Calculation of L10-Disorder Phase Boundary in Fe-Pd System, Mater. Trans., 2004, 45, p 1478-1484
T. Mohri and Y. Chen, First-Principles Investigation of L10-Disorder Phase Equilibria of Fe-Ni, -Pd, and -Pt Binary Alloy Systems, J. Alloys Compd., 2004, 383, p 23-31
J.W. Connolly and A.R. Williams, Density-Functional Theory Applied to Phase-Transformations in Transition-Metal Alloys, Phys. Rev. B, 1983, 27, p 5169-5172
J.M. Sanchez and D. de Fontaine, The Fee Ising Model in the Cluster Variation Approximation, Phys. Rev. B, 1978, 17, p 2926-2936
J.M. Sanchez, F. Ducastelle, and D. Gratias, Generalized Cluster Description of Multicomponent Systems, Physica (Utrecht), 1984, 128A, p 334-350
T. Mohri, J.M. Sanchez, and D. de Fontaine, Short-Range Order Diffuse Intensity Calculations in the Cluster Variation Method, Acta Metall., 1985, 33, p 1463-1474
W.L. Bragg and E.J. Williams, The Effect of Thermal Agitation on Atomic Arrangement in Alloys, Proc. R. Soc. Lond. A, 1934, 145, p 699-730
Acknowledgment
The present work was partly supported by Next Generation Supercomputing Project, Nanoscience Program, MEXT, Japan.
Author information
Authors and Affiliations
Corresponding author
Additional information
This article is an invited paper selected from participants of the 14th National Conference and Multilateral Symposium on Phase Diagrams and Materials Design in honor of Prof. Zhanpeng Jin’s 70th birthday, held November 3-5, 2008, in Changsha, China. The conference was organized by the Phase Diagrams Committee of the Chinese Physical Society with Drs. Huashan Liu and Libin Liu as the key organizers. Publication in Journal of Phase Equilibria and Diffusion was organized by J.-C. Zhao, The Ohio State University; Yong Du, Central South University; and Qing Chen, Thermo-Calc Software AB.
Rights and permissions
About this article
Cite this article
Mohri, T., Morita, T., Kiyokane, N. et al. Theoretical Investigation of Lattice Thermal Vibration Effects on Phase Equilibria Within Cluster Variation Method. J. Phase Equilib. Diffus. 30, 553–558 (2009). https://doi.org/10.1007/s11669-009-9571-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11669-009-9571-5