Thermochemistry of Alloys pp 1-28 | Cite as

# Modelling and Theories of Alloy Phase Behavior

## Abstract

It has been recognized since the early work of Hume-Rothery and others that many trends in alloy phase formation are readily understood in terms of physically plausible atomic parameters. For example, a substitutional alloy can only occur if there is not too great a difference in the sizes of the alloy constituents, so as not to require too great a cost in the elastic energy associated with deforming the lattice. This has led, in turn, to the introduction of so-called structural maps where two (or more) such atomic parameters are employed as the coordinates and well defined regions are observed to be associated with particular crystalline phases. These coordinates sometimes involve the difference in atomic parameters, such as the difference in the sizes of the constituent atoms, and sometimes involve an average, such as the average d-band occupancy of constituent transition element metals. An alternative approach to the emphasis on atomic parameters has been the consideration, as pioneered by Pearson, of how atoms are packed in some crystal structure and how this controls what the constituent atoms may be. Recently this has led to the utilization of Wigner-Seitz (sometimes called Voronoi or Dirichlet) constructs of the atomic cells in a crystal structure and, in turn, to the observation that sometimes two crystals which are nominally considered to have the same crystal structure according to normal crystallographic designation should, in fact, be considered to be different. The Wigner-Seitz cell constructs have also offered a framework for understanding trends in the magnetic and chemical properties of particular phases as well as making coordination between crystalline and glassy structures. Neither of the above approaches—correlations with atomic parameters or with packing considerations—provide numerical estimates of quantities of thermodynamic interest such as heats of formation. Such heats are being calculated with varying rigor and varying computational complexity ranging from model Hamiltonians employing atomic parameters to intricate electron band theory calculations. This chapter will attempt to provide the reader with a sense of some of successes and some of the problems when employing the above approaches to trace out trends in alloy phase behavior. Because of space limitations, this review will be highly selective.

## Keywords

Bond Line Atomic Parameter Interstitial Region Band Theory Total Energy Calculation## Preview

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## References

- 1.R. E. Watson and L. H. Bennett, ‘Alpha Manganese and the Frank-Kasper Phases’, Scripta Metall.
**19**, 535 (1985).CrossRefGoogle Scholar - 2.L. Brewer and R. H. Lamoreaux, ‘Phase Diagrams’, in Molybdenum: Physico-Chemical Properties of Its Compounds and Alloys, L. Brewer, Ed. (Int. Atomic Energy Agency, Vienna, 1980).Google Scholar
- 3.J. T. Waber, K. Gschneidner, Jr., A. C. Larson and M. Y. Price, ‘Prediction of Solid Solubility in Metallic Alloys’, Trans. Metall. Soc. AIME
**227**, 717 (1963).Google Scholar - 4.J. M. Lopez and J. A. Alonso ‘A Comparison of Two Parameterizations of Solid Solubility in Alloys: Thermochemical Coordinates Versus Orbital Radii Coordinates’, Physica
**113B**, 103 (1982).Google Scholar - 5.B. C. Giessen and S. H. Whang, ‘Metallic Glass Formation Diagrams’ in Alloy Phase Diagrams, L. H. Bennett, T. B. Massalski and B. C. Giessen, Eds., Materials Research Society Symposia Proceedings 19 (North Holland, NY, 1983).Google Scholar
- 6.E. N. Kaufman, R. Viarden, J. R. Chelikowsky and J. C. Phillips, ‘Extension of Equilibrium Formation Criteria to Metastable Microalloys’, Phys. Rev. Lett.
**39**, 1671 (1977).CrossRefGoogle Scholar - 7.e.g. A. R. Miedema, ‘On the Heat of Formation of Solid Alloys II’, J. Less-Common Metall.
**46**, 67 (1976).CrossRefGoogle Scholar - 8.J. St. John and A. N. Bloch, ‘Quantum-Defect Electronegativity Scale for Nontransition Elements’, Phys. Rev. Lett.
**33**, 1095 (1974).CrossRefGoogle Scholar - 9.R. E. Watson and L. H. Bennett, ‘Model Predictions of Volume Contractions in Transition-Metal Alloys and Implications for Laves Phase Formation’, Acta. Metall.
**32**, 491 (1984).CrossRefGoogle Scholar - 10.R. E. Watson and L. H. Bennett, ‘Transition-Metal Alloy Formation: The Occurrence of Topologically Closed-Packed Phases’, Acta. Metall.
**32**, 477 (1984).CrossRefGoogle Scholar - 11.D. G. Pettifor, ‘New Alloys from the Quantum Engineer’, New Scientist, May 29, 1986, p. 48.Google Scholar
- 12.P. Villars and L. D. Calvert, Pearson’s Handbook of Crystallographic Data for Intermetallic Phases (ASM, Metals Park, OH, 1985).Google Scholar
- 13.e.g. R. E. Watson and L. H. Bennett, ‘Electron Factors in the Occurrence of Sigma and Structurally Related Transition Metal Alloy Phases’, Scripta Metall.
**12**, 1165 (1978) and R. E. Watson and L. H. Bennett, ‘Transition Metals: d-Band Hybridization, Electronegativities and Structural Stability of Intermetallic Compounds’, Phys. Rev.**B18**, 6439 (1978).CrossRefGoogle Scholar - 14.P. Villars, J. C. Phillips and H. S. Chen, ‘Icosahedral Quasicrystals and Quantum Structure Diagrams’, Phys. Rev. Lett.
**57**, 3085 (1986).CrossRefGoogle Scholar - 15.W. P. Pearson, The Crystal Chemistry and Physics of Metals and Alloys, (Wiley, NY, 1972).Google Scholar
- 16.F. C. Frank and J. S. Kasper, ‘Complex Alloy Structures Regarded as Sphere Packings’, Acta Crystallogr.
**11**, 184 (1958) and**12**, 483 (1959).CrossRefGoogle Scholar - 17.W. Fischer, E. Koch and E. Hellner, ‘Zur Berechnung von Wirkungsbereichen In Strukturn Arorganlscher Uerblndungen’, N. Jb. Min. Mh. p. 227 (1971).Google Scholar
- 18.Also see, B. J. Gellatly and J. L. Finney, ‘Characterization of Models of Multicomponent Amorphous Metals: The Radical Alternative to the Voronoi Polyhedron’, J. Non-Crystall. Solids
**50**, 313 (1982).CrossRefGoogle Scholar - 19.W. B. Pearson, ‘Calculation in Establishing the Building Principles of the Crystal Structures of Intermetallic Phases’ in Computer Modeling of Phase Diagrams, L. H. Bennett, Ed. (The Metallurgical Soc., Warrendale, PA, 1986).Google Scholar
- 20.R. E. Watson and L. H. Bennett, ‘Disorderly Crystal Structures in Transition Metal Rich-Metalloid Alloys: Implications for Glass Formation’, Scripta Metall.
**17**, 827 (1983).CrossRefGoogle Scholar - 21.R. E. Watson and L. H. Bennett, ‘The Quasicrystalline Structures of Transition Metal/Metalloid Glasses’, J. Magn. Magnetic Mater.
**54–57**, 295 (1986).CrossRefGoogle Scholar - 22.D. R. Nelson, ‘Order, Frustration and Defects in Liquids and Glasses’, Phys. Rev.
**B28**, 5515 (1983).Google Scholar - 23.L. H. Bennett and R. E. Watson, ‘Symmetry and Supersymmetry in Crystals’, Phys. Rev.
**B35**, 845 (1987).Google Scholar - 24.L. H. Bennett, R. E. Watson and W. B. Pearson, ‘Topology of Local Atomic Environments: Implications for Magnetism and Superconductivity’, J. Magn. Magnetic Mater.
**54–57**, 1537 (1986).CrossRefGoogle Scholar - 25.R. E. Watson, M. Melamud and L. H. Bennett, ‘Disclinations and Magnetism in Rare-Earth-Transition-Metal Hard Magnets’, J. Appl. Phys.
**61**, 3580 (1987).CrossRefGoogle Scholar - 26.M. Melamud, L. H. Bennett and R. E. Watson, ‘Disclinations: Their Relation to the Anisotropies of Rare-Earth Hard Magnets’, Scripta Metall.
**21**, 573 (1987).CrossRefGoogle Scholar - 27.L. H. Bennett, A. J. McAlister and R. E. Watson, ‘Interstitial Compounds’, Physics Today
**30**, 34 (1977).CrossRefGoogle Scholar - 28.D. G. Pettifor, ‘Theory of the Heats of Formation of Transition-Metal Alloys’, Phys. Rev. Lett.
**42**, 846 (1979).CrossRefGoogle Scholar - 29.C. M. Varma, ‘Quantum Theory of the Heats of Formation of Metallic Alloys’, Solid State Commun.
**31**, 295 (1979).CrossRefGoogle Scholar - 30.R. E. Watson and L. H. Bennett, ‘Optimized Prediction for Heats of Formation of Transition-Metal Alloys II’, CALPHAD
**8**, 307 (1984).CrossRefGoogle Scholar - 31.e.g. Theory of the Inhomogeneous Electron Gas, S. Lundqvist and N. H. March, Eds. (Plenum, NY, 1983).Google Scholar
- 32.R. E. Watson, J. W. Davenport and M. Weinert, ‘Linear Augmented Slater-type Orbital Study of Au-5d Transition Metal Alloying’, Phys. Rev.
**B35**, 508 (1987).Google Scholar - 33.Phase Stablllllty in Metals and Alloys, P. S. Rudman, J. Stringer and R. I. Jaffee, Eds. (McGraw-Hill, NY (1967).Google Scholar
- 34.L. Kaufman and H. Bernstein, Computer Calculation of Phase Diagrams (Academic Press, NY, 1970).Google Scholar
- 35.H. L. Skriver, ‘Crystal Structure from One-electron Theory’, Phys. Rev.
**B32**, 1909 (1985).Google Scholar - 36.A. P. Mlodownlk, ‘The Phase Stability of the Elements’, in Computer Modeling of Phase Diagrams, L. H. Bennett, Ed. (The Metallurgical Soc., Warrendale, PA, 1986).Google Scholar
- 37.G. Grimvall, M. Thiessen and A. F. Guillermet, ‘Thermodynamic Properties of Tungsten’, Phys. Rev. (to appear).Google Scholar
- 38.R. E. Watson, J. W. Davenport and M. Weinert, ‘A Linear Augmented Slater-type Orbital Study of Pt-5d Transition Metal Alloying’, Phys. Rev. (to appear).Google Scholar
- 39.For a recent example see K. J. Chang and M. L. Cohen ‘Ab initio Psuedopotential Study of Structural and High Pressure Properties of SiC’, Phys. Rev.
**B35**, 8196 (1987).Google Scholar - 40.For a recent example see C. Sigli, M. Kosugi and M. M. Sanchez, ‘Calculation of Thermodynamic Properties and Phase Diagrams of Binary Transition Metal Alloys’, Phys. Rev. Lett.
**57**, 253 (1986).CrossRefGoogle Scholar - 41.L. F. Mattheiss and D. R. Hamann, ‘Linear Augmented-Plane-Wave Calculation of the Structural Properties of Bulk Cr, Mo and W’, Phys. Rev.
**33B**, 823 (1986).Google Scholar - 42.L. F. Mattheiss and D. R. Hamann, ‘Electronic Charge Density of V
_{3}Si’, Solid State Commun.**38**, 689 (1981).CrossRefGoogle Scholar