Surface Rate Limitations and Segregation

  • Richard Ghez
Chapter

Abstract

The purpose in this chapter is to clarify the notion of “local equilibrium,” namely, when can we assert that the concentration at a phase boundary is pinned to its equilibrium value, even as the core of an adjacent bulk phase undergoes nonequilibrium processes. To that end, we must first explain what is a “phase boundary, ” and what does “concentration at a phase boundary” really mean. These questions are of immense practical importance for such applications as the change in catalytic properties of metal surfaces over time and the change in doping character of semiconductors during annealing. They are all closely related to the question of surface segregation to free surfaces and to grain boundaries, which are topics of scientific interest in their own right.1–3

Keywords

Phase Boundary Surface Concentration Concentration Distribution Boundary Diffusion Diffusion Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. McLean, Grain Boundaries in Metals (Oxford University Press, London, 1957).Google Scholar
  2. 2.
    Interfacial Segregation, W. C. Johnson and J. M. Blakely, Eds. (Amer. Soc. for Metals, Ohio 1979).Google Scholar
  3. 3.
    J. Cabané and F. Cabané, “Equilibrium Segregation in Interfaces, ” in Interface Segregation and Related Processes in Materials, J. Novotny, Ed., pp. 1–160 (Trans. Tech. Publ., Brookfield, 1991).Google Scholar
  4. 4.
    H. S. Carslaw, Introduction to the Theory of Fourier’s Series and Integrals, 3rd ed. Macmillan, New York, 1930. (Reprinted by Dover, New York).MATHGoogle Scholar
  5. 5.
    E. Isaacson and H. B. Keller, Analysis of Numerical Methods (John Wiley, New York, 1966).MATHGoogle Scholar
  6. 6.
    R. H. Eddy and R. Fritsch, “An Optimization Oddity, ” College Mathematics J. 25, 227 (1994).CrossRefMATHGoogle Scholar
  7. 7.
    R. Paré, “A Visual Proof of Eddy and Fritsch’s Minimal Area Property, ” College Mathematics J. 26, 43 (1995).CrossRefMATHGoogle Scholar
  8. 8.
    J. W. Gibbs, “On the Equilibrium of Heterogeneous Substances, ” Trans. Conn. Acad. (1875–1878). [Reprinted in Scientific Papers, Vol. 1 (Dover, New York, 1961).]Google Scholar
  9. 9.
    J. D. van der Waals, “The Thermodynamic Theory of Capillarity under the Hypothesis of a Continuous Variation of Density, ” originally published in Dutch (1893); translated by J. S. Rowlinson in J. Statist. Phys. 20, 197 (1979).Google Scholar
  10. 10.
    J. W. Cahn and J. E. Hilliard, “Free Energy of a Nonuniform System, ” Parts I-III, J. Chem. Phys. 28, 258 (1958)ADSCrossRefGoogle Scholar
  11. 10a.
    J. W. Cahn and J. E. Hilliard, “Free Energy of a Nonuniform System, ” Parts I-III, J. Chem. Phys. 30, 1121 (1959)ADSCrossRefGoogle Scholar
  12. 10b.
    J. W. Cahn and J. E. Hilliard, “Free Energy of a Nonuniform System, ” Parts I-III, J. Chem. Phys. 31, 688 (1959).ADSCrossRefGoogle Scholar
  13. 11.
    L. Prandtl, “Über Flüssigkeitsbewegung bei sehr kleiner Reibung, ” Proceedings 3rd Intern. Math. Congr., Heidelberg (1904), pp. 484–491. [See Collected Works, Vol. 2, pp. 575–584.]MATHGoogle Scholar
  14. 12.
    H. Schlichting, “Boundary-Layer Theory, ” 6th ed. (McGraw-Hill, New York, 1968).MATHGoogle Scholar
  15. 13.
    M. Van Dyke, Perturbation Methods in Fluid Mechanics, 2nd annotated edition (The Parabolic Press, Stanford, CA, 1975).MATHGoogle Scholar
  16. 14.
    H. J. Queisser, K. Hubner, and W. Shockley, “Diffusion along Small-Angle Grain Boundaries in Silicon, ” Phys. Rev. 123, 1245 (1961).ADSCrossRefGoogle Scholar
  17. 15.
    C. Lea and M. P. Seah, “Kinetics of Surface Segregation, ” Phil. Mag. A35, 213 (1977).CrossRefGoogle Scholar
  18. 16.
    W. R. Tyson, “Kinetics of Temper Embrittlement, ” Acta Met. 26, 1471 (1978).CrossRefGoogle Scholar
  19. 17.
    R. T. P. Whipple, “Concentration Contours in Grain Boundary Diffusion, ” Phil. Mag. 45, 1225 (1954).CrossRefMATHGoogle Scholar
  20. 18.
    A. D. Brailsford, “Surface Segregation Kinetics in Binary Alloys, ” Surf Sci. 94, 387 (1980).ADSCrossRefGoogle Scholar
  21. 19.
    P. Benoist and G. Martin, “Atomic Model for Grain Boundary and Surface Diffusion, ” Thin Solid Films 25, 181 (1975)ADSCrossRefGoogle Scholar
  22. 19a.
    P. Benoist and G. Martin, “Atomic Model for Grain Boundary and Surface Diffusion, ” errata, 27, L8 (1975).Google Scholar
  23. 20.
    G. Rowlands and D. P. Woodruff, “The Kinetics of Surface and Grain Boundary Segregation in Binary and Ternary Systems, ” Phil. Mag. A40, 459 (1979).ADSCrossRefGoogle Scholar
  24. 21.
    R. Ghez, “Irreversible Thermodynamics of a Stationary Interface, ” Surf. Sci. 20, 326 (1970).ADSCrossRefGoogle Scholar
  25. 22.
    D. A. Edwards, H. Brenner, and D. T. Wasan, Interface Transport Processes and Rheology, (Butterworth-Heinemann, Boston, 1991).Google Scholar
  26. 23.
    S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1962). [Reprinted by Dover, New York.]MATHGoogle Scholar
  27. 24.
    H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford University Press, London, 1959).MATHGoogle Scholar
  28. 25.
    W. H. Reinmuth, “Diffusion to a Plane with Langmuirian Adsorption, ” J. Phys. Chem. 65, 473 (1961).CrossRefGoogle Scholar
  29. 26.
    R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena (John Wiley, New York, 1960).Google Scholar
  30. 27.
    G. H. Gilmer, R. Ghez, and N. Cabrera, “An Analysis of Combined Surface and Volume Diffusion Processes in Crystal Growth, ” Parts I and II, J. Cryst. Growth 8, 79 (1971)ADSCrossRefGoogle Scholar
  31. 27a.
    G. H. Gilmer, R. Ghez, and N. Cabrera, “An Analysis of Combined Surface and Volume Diffusion Processes in Crystal Growth, ” Parts I and II, J. Cryst. Growth 21, 93 (1974).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Richard Ghez

There are no affiliations available

Personalised recommendations