On the motion of a phase interface by surface diffusion

  • Fabrizio Davì
  • Morton E. Gurtin
Original Papers


Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullins's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.


Free Energy Alla Mathematical Method General Framework Capillary Force 


Mullins, in una serie di articoli inerenti la morfologia delle superfici di interfaccia tra fasi, ha sviluppato una dinamica delle superfici la cui evoluzione è governata dal fenomeno di diffusione di massa all'interno dell'interfaccia. Scopo di questo articolo è inscrire la teoria di Mullins in uno schema più generale basato su leggi di bilancio della massa e delle azioni capillari nonchè su una formulazione puramente meccanica del secondo principio della termodinamica, asserente ehe l'incremento di energia libera non possa essere superiore al flusso di energia ed alla potenza fornite all'interfaccia. Viene successivamente sviluppata una appropriata teoria costitutiva, e vengono dedotte le equazioni di evoluzione sia in forma generale che approssimata.


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  1. [AC]
    Allen, S. M. and J. W. Cahn,A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Act. Metall.27, 1085–1098 (1979).Google Scholar
  2. [AG]
    Angenent, S. and M. E. Gurtin,Multiphase thermomechanics with interfacial structure. 2. Evolution of an isothermal interface, Arch. Rational Mech. Anal.108, 323–391 (1989).Google Scholar
  3. [Br]
    Brakke, K. A.,The Motion of a Surface by its Mean Curvature, Princeton University Press, 1978.Google Scholar
  4. [BCF]
    Burton, W. K., Cabrera, N., and F. C. Frank,The growth of crystals and the equilibrium structure of their surfaces, Phil. Trans. Roy. Soc. LondonA243, 299–358 (1951).Google Scholar
  5. [CGG]
    Chen, Y. G., Giga, Y., and S. Goto,Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Forthcoming.Google Scholar
  6. [ES]
    Evans, L. C. and J. Spruck,Motion of level sets by mean curvature I, Forthcoming.Google Scholar
  7. [GH]
    Gage, M. and R. S. Hamilton,The heat equation shrinking convex plane curves, J. Diff. Geom.23, 69–95 (1986).Google Scholar
  8. [Gr]
    Grayson, M. A.,The heat equation shrinks embedded plane curves to round points, J. Diff. Geom.26, 285–314 (1987).Google Scholar
  9. [G1]
    Gurtin, M. E.,Multiphase thermomechanics with interfacial structure. 1. Heat conduction and the capillary balance law, Arch. Rational Mech. Anal.104, 195–221 (1988).Google Scholar
  10. [G2]
    Gurtin, M. E.,On the two-phase Stefan problem with interfacial energy and entropy, Arch. Rational Mech. Anal.96, 199–241 (1986).Google Scholar
  11. [G3]
    Gurtin, M. E.,On thermomechanical laws for the motion of a phase interface, Forthcoming.Google Scholar
  12. [GPG]
    Gurtin, M. E. and P. Podio Guidugli,A hyperbolic theory for the evolution of plane curves, SIAM J. Math. Anal. Forthcoming.Google Scholar
  13. [GS]
    Gurtin, M. E. and A. Struthers,Multiphase thermomechanics with interfacial structure. 3. Evolving phase boundaries in the presence of bulk deformation, Arch. Rational Mech. Anal. Forthcoming.Google Scholar
  14. [GSW]
    Gurtin, M. E., Struthers, A., and W. O. Williams,A transport theorem for moving interfaces, Quart. Appl. Math. Forthcoming.Google Scholar
  15. [He]
    Herring, C.,Surface tension as a motivation for sintering. The Physics of Powder Metallurgy (ed. W. E. Kingston) McGraw-Hill, New York 1951.Google Scholar
  16. [M1]
    Mullins, W. W.,Two-dimensional motion of idealized grain boundaries, J. Appl. Phys.,27, 900–904 (1956).Google Scholar
  17. [M2]
    Mullins, W. W.,Theory of thermal grooving, J. Appl. Phys.28, 333–339 (1957).Google Scholar
  18. [M3]
    Mullins, W. W.,The effect of thermal grooving on grain boundary motion, Acta Met.6, 414–427 (1958).Google Scholar
  19. [M4]
    Mullins, W. W.,Grain boundary grooving by volume diffusion, Trans. Met. Soc. AIME218, 354–361 (1960).Google Scholar
  20. [M5]
    Mullins, W. W.,Theory of linear facet growth during thermal etching, Phil. Mag.6, 1313–1341 (1961).Google Scholar
  21. [M6]
    Mullins, W. W.,Solid surface morphologies governed by capillarity, Metal Surfaces: Structure, Energetics, and Kinetics, Am. Soc. Met.17, 17–66, Cleveland (1961).Google Scholar
  22. [NM]
    Nichols, F. A. and W. W. Mullins,Morphological changes of a surface of revolution due to capillarity-induced surface diffusion, J. Appl. Phys.36, 1826–1835 (1965).Google Scholar
  23. [OS]
    Osher, S. and J. A. Sethian,Front propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys.79, 12–49 (1988).Google Scholar
  24. [Se]
    Sethian, J. A.,Curvature and the evolution of fronts, Comm. Math. Phys.101, 487–499 (1985).Google Scholar

Copyright information

© Birkhäuser Verlag 1990

Authors and Affiliations

  • Fabrizio Davì
    • 1
  • Morton E. Gurtin
    • 2
  1. 1.Dipartimento di Ingegneria Civile EdileUniversità di Roma 2RomaItaly
  2. 2.Dept of MathematicsCarnegie-Mellon UniversityPittsburghUSA

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