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A General Method of Solution for the Cluster Variation Method in Ionic Solids, with Application to Diffusionless Transitions in Yttria-Stabilized Zirconia

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Abstract

A new, general method of solution for the cluster variation method using a reduced conjugate gradient approach with a truncated line-search algorithm is presented. The method is generally convergent. Additionally, the truncation of the line-search algorithm may increase the speed of convergence considerably, as the size of the problem is progressively reduced (especially for strongly ordered phases), opening up the possibility of a considerable increase in the size of maximal clusters. The method is successfully demonstrated for a single, eight-atom maximal cluster in the fluorite lattice. Using pairwise defect interaction energies calculated for cubic, yttria-doped zirconia and fixed defect concentrations, a pair of metastable states are found in a composition and temperature range which is experimentally characterized by metastable, diffusionless phase transitions.

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References

  1. An, G.Z.: A note on the cluster variation method. J. Stat. Phys. 52(3–4), 727–734 (1988)

    Article  MATH  ADS  Google Scholar 

  2. Barker, J.A.: Methods of approximation in the theory of regular mixtures. Proc. R. Soc. A 216(1124), 45–56 (1953)

    Article  ADS  Google Scholar 

  3. Blöchl, P.E.: Projector augmented-wave method. Phys. Rev. B 50(24), 17953–17979 (1994)

    Article  ADS  Google Scholar 

  4. Bogicevic, A., Wolverton, C.: Nature and strength of defect interactions in cubic stabilized zirconia. Phys. Rev. B 67(2), 024106 (2003)

    Article  ADS  Google Scholar 

  5. Bogicevic, A., Wolverton, C., Crosbie, G.M., Stechel, E.B.: Defect ordering in aliovalently doped cubic zirconia from first principles. Phys. Rev. B 6401(1), 014106 (2001)

    Article  ADS  Google Scholar 

  6. Caracoche, M.C., Martinez, J.A., Rivas, P.C., Rodriguez, A.M., Lamas, D.G., Lascalea, G.E., de Reca, N.E.W.: Hyperfine characterization of the metastable t″-form of the tetragonal phase in ZrO2—10 mol % Y2O3 powders synthesized by gel combustion. J. Am. Ceram. Soc. 88(6), 1564–1567 (2005)

    Article  Google Scholar 

  7. Ceperley, D.M., Alder, B.J.: Ground-state of the electron-gas by a stochastic method. Phys. Rev. Lett. 45(7), 566–569 (1980)

    Article  ADS  Google Scholar 

  8. Chen, M., Hallstedt, B., Gauckler, L.J.: Thermodynamic modeling of the ZrO2–YO1.5 system. Solid State Ion. 170(3–4), 255–274 (2004)

    Article  Google Scholar 

  9. Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley-Interscience, New York (1987)

    MATH  Google Scholar 

  10. Goff, J.P., Hayes, W., Hull, S., Hutchings, M.T., Clausen, K.N.: Defect structure of yttria-stabilized zirconia and its influence on the ionic conductivity at elevated temperatures. Phys. Rev. B 59(22), 14202–14219 (1999)

    Article  ADS  Google Scholar 

  11. Heskes, T., Albers, K., Kappen, B.: Approximate inference and constrained optimization. In: M. Kaufmann (ed.) Proceedings of the 19th Annual Conference on Uncertainty in Artificial Intelligence (UAI-03), p. 313 (2003)

  12. Kelly, P.M., Rose, L.R.F.: The martensitic transformation in ceramics: its role in transformation toughening. Prog. Mater. Sci. 47(5), 463–557 (2002)

    Article  Google Scholar 

  13. Kikuchi, R.: A theory of cooperative phenomena. Phys. Rev. 81(6), 988–1003 (1951)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Kikuchi, R.: Natural iteration method and boundary free-energy. J. Chem. Phys. 65(11), 4545–4553 (1976)

    Article  ADS  Google Scholar 

  15. Kikuchi, R., Brush, S.G.: Improvement of cluster-variation method. J. Chem. Phys. 47(1), 195 (1967)

    Article  ADS  Google Scholar 

  16. Kresse, G., Hafner, J.: Ab-initio molecular-dynamics for liquid-metals. Phys. Rev. B 47(1), 558–561 (1993)

    Article  ADS  Google Scholar 

  17. Kresse, G., Hafner, J.: Ab-initio molecular-dynamics simulation of the liquid-metal amorphous-semiconductor transition in germanium. Phys. Rev. B 49(20), 14251–14269 (1994)

    Article  ADS  Google Scholar 

  18. Kresse, G., Joubert, D.: From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59(3), 1758–1775 (1999)

    Article  ADS  Google Scholar 

  19. Morita, T.: General structure of distribution functions for Heisenberg model and Ising-model. J. Math. Phys. 13(1), 115 (1972)

    Article  ADS  Google Scholar 

  20. Nakayama, M., Martin, M.: First-principles study on defect chemistry and migration of oxide ions in ceria doped with rare-earth cations. Phys. Chem. Chem. Phys. 11(17), 3241–3249 (2009)

    Article  Google Scholar 

  21. Pelizzola, A.: Cluster variation method in statistical physics and probabilistic graphical models. J. Phys. A 38(33), R309–R339 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  22. Perdew, J.P., Wang, Y.: Accurate and simple analytic representation of the electron-gas correlation-energy. Phys. Rev. B 45(23), 13244–13249 (1992)

    Article  ADS  Google Scholar 

  23. Pornprasertsuk, R., Ramanarayanan, P., Musgrave, C.B., Prinz, F.B.: Predicting ionic conductivity of solid oxide fuel cell electrolyte from first principles. J. Appl. Phys. 98(10), 103513 (2005)

    Article  ADS  Google Scholar 

  24. Predith, A., Ceder, G., Wolverton, C., Persson, K., Mueller, T.: Ab initio prediction of ordered ground-state structures in ZrO2–Y2O3. Phys. Rev. B 77(14), 144104 (2008)

    Article  ADS  Google Scholar 

  25. Pretti, M.: On the convergence of Kikuchi’s natural iteration method. J. Stat. Phys. 119(3-4), 659–675 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Rodriguez, A.M., Caracoche, M.C., Rivas, P.C., Pasquevich, A.F., Mintzer, S.R.: PAC characterization of nontransformable tetragonal t′ phase in arc-melted zirconia—2.8 mol % yttria ceramics. J. Am. Ceram. Soc. 84(1), 188–192 (2001)

    Article  Google Scholar 

  27. Ruszczynski, A.: Nonlinear Optimization. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  28. Sanchez, J.M., de Fontaine, D.D.: FCC Ising-model in cluster variation approximation. Phys. Rev. B 17(7), 2926–2936 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  29. Schlijper, A.: Convergence of the cluster-variation method in the thermodynamic limit. Phys. Rev. B 27(11), 6841–6848 (1983)

    Article  ADS  Google Scholar 

  30. Tepesch, P.D., Garbulsky, G.D., Ceder, G.: Model for configurational thermodynamics in ionic systems. Phys. Rev. Lett. 74(12), 2272–2275 (1995)

    Article  ADS  Google Scholar 

  31. Veldhuizen, T.L.: Arrays in Blitz++. In: Computing in Object-Oriented Parallel Environments. Second International Symposium, ISCOPE 98, pp. 223 (1998)

  32. Yashima, M., Kakihana, M., Yoshimura, M.: Metastable-stable phase diagrams in the zirconia-containing systems utilized in solid-oxide fuel cell application. Solid State Ion. 86-8, 1131–1149 (1996)

    Article  Google Scholar 

  33. Yashima, M., Sasaki, S., Kakihana, M., Yamaguchi, Y., Arashi, H., Yoshimura, M.: Oxygen-induced structural-change of the tetragonal phase around the tetragonal-cubic phase-boundary in ZrO2–YO1.5 solid-solutions. Acta Cryst. 50, 663–672 (1994)

    Article  Google Scholar 

  34. Yuille, A.L.: CCCP algorithms to minimize the Bethe and Kikuchi free energies: Convergent alternatives to belief propagation. Neural Comput. 14(7), 1691–1722 (2002)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569, Stuttgart, Germany

    D. S. Mebane

  2. Department of Chemistry, National Taiwan Normal University, Taipei, Taiwan

    J. H. Wang

Authors
  1. D. S. Mebane
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  2. J. H. Wang
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Correspondence to D. S. Mebane.

Additional information

Support for D.S. Mebane provided by the National Science Foundation International Research Fellowship Program, Grant No. 0701145.

This work was further supported by DOE Basic Energy Sciences under Grant No. DE-FG02-06ER15837.

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Mebane, D.S., Wang, J.H. A General Method of Solution for the Cluster Variation Method in Ionic Solids, with Application to Diffusionless Transitions in Yttria-Stabilized Zirconia. J Stat Phys 139, 727–742 (2010). https://doi.org/10.1007/s10955-010-9963-2

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  • DOI: https://doi.org/10.1007/s10955-010-9963-2

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