Skip to main content
SpringerLink
Log in

Theories of Coherent Phase Equilibrium

  • Chapter
  • First Online:
The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components

Part of the book series: Engineering Materials ((ENG.MAT.))

  • 1356 Accesses

Abstract

A general treatment of phase equilibrium of binary substitutional, or interstitial (metal-hydrogen) coherent solids is given in this chapter. In such systems, the total coherency energy between the misfitting species and/or phases includes a contribution from the indirect (non-configurational) interactions that occur in solids of finite size between the misfitting phases or atomic species. It is shown that, as a result of this latter contribution and the associated dependence of the total coherency energy on the square of the concentrations of the misfitting phases or atomic species in the solid, the equilibrium phase relationships for such solids differ significantly from the classical, Gibbsian thermodynamic equilibrium relationships applicable to liquid and hydrostatically stressed (incoherent) solid systems. Two bounding cases of the application of these coherency energy models are reviewed in this chapter.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    It is not possible with this method to determine whether the state obtained represents an absolute minimum or not.

References

  1. Cahn, J.W.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961)

    Article  Google Scholar 

  2. Cahn, J.W.: On spinodal decomposition in cubic crystals. Acta Metall. 10, 179–183 (1962)

    Article  Google Scholar 

  3. Cahn, J.W.: Coherent fluctuations and nucleation in isotropic solids. Acta Metall. 10, 907–913 (1962)

    Article  Google Scholar 

  4. Cahn, J.W., Larché, F.C.: A simple model for coherent equilibrium. Acta Metall. 32, 1915–1923 (1984)

    Article  Google Scholar 

  5. Chiang, C.S., Johnson, W.C.: Coherent phase equilibria in systems possessing a consolute critical point. J. Mater. Res. 4, 678–687 (1989)

    Article  Google Scholar 

  6. Eshelby, J.S.: Continuum theory of defects. Prog. Solid State Phys. 3, 79–144 (1956)

    Article  Google Scholar 

  7. Eshelby, J.D.: Interaction and diffusion of point defects. In: Smallman, R.E.(ed.) Vacancies 76, London, The Metals Society, pp. 3–10 (1976)

    Google Scholar 

  8. Everett, D.H., Norton, P.: Hysteresis in the Palladium-Hydrogen System. Proc. Roy. Soc. A 259, 341–360 (1960)

    Google Scholar 

  9. Flanagan, T.B., Clewley, T., Kuji, T et al.: Isobaric and isothermal hysteresis in metal hydrides and oxides. J. Chem. Soc., Faraday Trans. 1. 82, 2589–2604 (1986)

    Google Scholar 

  10. Flanagan, T.B., Park, C.-N., Oates, W.A.: Hysteresis in solid state reactions. Prog. Solid St. Chem. 23, 291–363 (1995)

    Article  Google Scholar 

  11. Hillert, M.: In: Phase Equilibrium, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis, 2nd edn. Cambridge University Press, Cambridge, UK (2008)

    Google Scholar 

  12. Johnson, W.C., Voorhees, P.W.: Phase equilibrium in two-phase coherent solids. Metall. Trans. A 18A, 1213–1228 (1987)

    Google Scholar 

  13. Johnson, W.C., Müller, W.H.: Characteristics of phase equilibria in coherent solids. Acta Metall. Mater. 39, 89–103 (1991)

    Article  Google Scholar 

  14. Larché, F.C., Cahn, J.W.: A linear theory of thermochemical equilibrium of solids under stress. Acta Metall. 21, 1051–1063 (1973)

    Article  Google Scholar 

  15. Larché, F.C., Cahn, J.W.: A nonlinear theory of thermochemical equilibrium of solids under stress. Acta Metall. 26, 53–60 (1978)

    Article  Google Scholar 

  16. Larché, F.C., Cahn, J.W.: Thermochemical equilibrium of multiphase solids under stress. Acta Metall. 26, 1579–1589 (1978)

    Article  Google Scholar 

  17. Larché, F.C., Cahn, J.W.: The effect of self-stress on diffusion in solids. Acta Metall. 30, 1835–1845 (1982)

    Article  Google Scholar 

  18. Larché, F.C., Cahn, J.W.: The interactions of composition and stress in crystalline solids. Acta Metall. 33, 331–357 (1985)

    Article  Google Scholar 

  19. Lee, J.K., Tao, W.: Coherent phase equilibria: effect of composition-dependent elastic strain. Acta Metall. Mater. 42, 569–577 (1994)

    Article  Google Scholar 

  20. Liu, Z.-K., Ågren, J.: On two-phase coherent equilibrium in binary alloys. Acta Metall. Mater. 38, 561–572 (1990)

    Article  Google Scholar 

  21. Liu, Z.-K., Ågren, J.: Two-phase coherent equilibrium in multicomponent alloys. J. Phase Equil. 12, 266–274 (1991)

    Article  Google Scholar 

  22. Pfeifer, M.J., Voorhees, P.W.: A graphical method for constructing coherent phase diagrams. Acta Metall. Mater. 39, 2001–2012 (1991)

    Article  Google Scholar 

  23. Roǐtburd, A.L.: Equilibrium and phase diagrams of coherent phases in solids. Sov. Phys.—Solid State 26, 1229–1233 (1984)

    Google Scholar 

  24. Schwarz, R.B., Khachaturyan, A.G.: Thermodynamics of open two-phase systems with coherent interfaces. Phys. Rev. Letters 74, 2523–2526 (1995)

    Article  Google Scholar 

  25. Schwarz, R.B., Khachaturyan, A.G.: Thermodynamics of open two-phase systems with coherent interfaces: application to metal-hydrogen systems. Acta Mater. 54, 313–323 (2006)

    Article  Google Scholar 

  26. Williams, R.O.: Long-period superlattices in the copper-gold system as two-phase mixtures. Metall. Trans. A 11A, 247–253 (1980)

    Google Scholar 

  27. Williams, R.O.: The calculation of coherent phase equilibria. CALPHAD 8, 1–14 (1984)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MPP Consulting, 1242 Donlea Crescent, Oakville, Ontario, L6J 1V7, Canada

    Manfred P. Puls

Authors
  1. Manfred P. Puls
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manfred P. Puls .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag London

About this chapter

Cite this chapter

Puls, M.P. (2012). Theories of Coherent Phase Equilibrium. In: The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components. Engineering Materials. Springer, London. https://doi.org/10.1007/978-1-4471-4195-2_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4471-4195-2_7

  • Published:

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-4194-5

  • Online ISBN: 978-1-4471-4195-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation