Advertisement

Curvature Dependent Phase Boundary Motion and Parabolic Double Obstacle Problems

  • J. F. Blowey
  • C. M. Elliott
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 47)

Abstract

The use of parabolic double obstacles problems for approximating curvature dependent phase boundary motion is reviewed. It is shown that such problems arise naturally in multi-component diffusion with capillarity. Formal matched asymptotic expansions are employed to show that phase field models with order parameter solving an obstacle problem approximate curvature dependent phase boundary motion. Numerical simulations of surfaces evolving according to their mean curvature are presented.

Keywords

Curvature Flow Spinodal Decomposition Stefan Problem Phase Field Model Surface Tension Effect 
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. S. M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall, 27 (1979), pp. 1085–1095.CrossRefGoogle Scholar
  2. H. W. Alt and I. Pawlow, A mathematical model of dynamics of non-isothermal phase separation, Technical report no. 158, Rheinische Friedrich-Wilhelms-Univerität, Bonn, Germany (1991).Google Scholar
  3. P. Bates & Songmu Zheng, Inertial manifold and inertial sets for phase-field equations, IMA preprint #809 University of Minnesota, Minneapolis, MN 55455, USA (1991).Google Scholar
  4. G. Bellettini, M. Paolini and C. Verdi, T-convergence of discrete approximations to interfaces with prescribed mean curvature, Rend. Atti. Naz. Lincei, 1 (1990), pp. 317–328.MathSciNetMATHGoogle Scholar
  5. J. F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy Part I: Mathematical analysis, Euro. Jnl. of Applied Mathematics, 2 (1991), pp. 233–279.MathSciNetMATHCrossRefGoogle Scholar
  6. J. F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy Part II: Numerical analysis, To appear in Euro. Jnl. of Applied Mathematics (1992).Google Scholar
  7. D. Brochet, Xinfu Chen: D. Hilhorst, Finite dimensional exponential attractor for the phase field model, IMA preprint #858 University of Minnesota, Minneapolis, MN 55455, USA (1991).Google Scholar
  8. L. Bronsard and R.V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Diff. Eqns. (1990), pp. 211–237.Google Scholar
  9. G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal, 92 (1986), pp. 205–245.MathSciNetMATHCrossRefGoogle Scholar
  10. G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of phase field equations, Phys. Rev. A., 39 (1989), pp. 887–896.MathSciNetCrossRefGoogle Scholar
  11. J. W. Cahn, On spinodal decomposition, Acta Metall, 9 (1961), pp. 795–801.CrossRefGoogle Scholar
  12. A. Cerezo, M. Hetherington, J. Hyde, G. Smith, M. Copetti and C.M. Elliott, Nano structures and the position sensitive atom-probe, ©University of Oxford (1990).Google Scholar
  13. J. W. Cahn and J.E. Hilliard, Free energy of a non-uniform system I. Interfacial free energy, J. Chem. Phys., 28 (1957), pp. 258–267.CrossRefGoogle Scholar
  14. Y. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. DifF. Geom., 33 (1991), pp. 749–786.MathSciNetMATHGoogle Scholar
  15. Xinfu Chen, Generation and propogation of interface in reaction-diffusion equations, J. Diff. Eqns. to appear (1992).Google Scholar
  16. Xinfu Chen and C.M. Elliott, Asymptotics for a parabolic double obstacle problem, Preprint submitted for publication.Google Scholar
  17. M. I. M. C. Copetti and C.M. Elliott, Numerical analysis of the Cahn-Hilliard equation with logarithmic free energy, Numerische Mathematik to appear 1992).Google Scholar
  18. P. De Mottoni and M. Schatzmann, Evolution géométrique d’interfaces, C.R. Acad. Sci. Sér. I math., 309 (1989), pp. 453–458.MATHGoogle Scholar
  19. C. M. Elliott and S. Luckhaus, A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy, Submitted for publication (1991).Google Scholar
  20. C. M. Elliott and J.R. Ockendon, Weak & variational methods for free and moving boundary problems, Pitman, London, 1982.Google Scholar
  21. C. M. Elliott and Songmu Zheng, Global existence and stability of solutions to the phase field equations, In International series of numerical methods 95 (eds. K.H. Hoffmann and J. Sprekels), Birkhäuser, Basel, 1990, 46–58.Google Scholar
  22. L. C. Evans, H.M. Soner and P.E. Souganidis, Phase transitions and generalized motion by mean curvature, To appear in Comm. Pure & Appl. Math (1992).Google Scholar
  23. L. C. EVANS and J. Spruck, Motion of level sets by mean curvature I, J. Diff. Geom., 33 (1991), pp. 635–681.MathSciNetMATHGoogle Scholar
  24. D. J. Eyre, Systems of Cahn-Hilliard equations, preprint (1991).Google Scholar
  25. P. C. Fife, Dynamics of internal layers and diffusive interfaces, SIAM, Philadelphia, 1988.CrossRefGoogle Scholar
  26. A. Friedman, Variational principles and free-boundary problems, Wiley-interscience, New York, 1982.MATHGoogle Scholar
  27. M. Gage and R.S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom., 23 (1986), pp. 69–96.MathSciNetMATHGoogle Scholar
  28. M. Grayson, The heat equation shrink embedded curves to round points, J. DifF. Geom., 26 (1987), pp. 285–314.MathSciNetMATHGoogle Scholar
  29. E. Guisti, Minimal surfaces and functions of bounded variation, Birkhäuser Verlag, 1984.Google Scholar
  30. R. S. Hamilton, Three manifolds with positive Ricci curvature, J. Diff. Geom., 26 (1982), pp. 255–306.MathSciNetGoogle Scholar
  31. J. J. Hoyt, Linear spinodal decomposition in a regular ternary alloy, Acta Metall., 38 (1990a), pp. 227–231.CrossRefGoogle Scholar
  32. J. J. Hoyt, The continuum theory of nucleation in multi-component systems, Acta Metall., 38 (1990b), pp.1405–1412.CrossRefGoogle Scholar
  33. G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Diff. Geom., 20 (1984), pp. 237–266.MathSciNetMATHGoogle Scholar
  34. P. C. Hohenberg and B.I. Halperin, Theory of dynamical critical phenomena, Rev. Mod. Phys., 49 (1977), pp. 435–479.CrossRefGoogle Scholar
  35. S. Luckhaus, Solutions for the two phase Stefan problem with the Gibbs-Thomson law for the melting temperature, Euro. Jnl. of Applied Mathematics, 1 (1990), pp. 101–111.MathSciNetMATHCrossRefGoogle Scholar
  36. S. Luckhaus, and L. Modica, The Gibbs-Thompson relation within the gradiet theory of phase transitions, Arch. Rat. Mech. Anal., 107(1) (1989), pp. 71–83.MathSciNetMATHCrossRefGoogle Scholar
  37. A. M. Meirmanov, The Stefan problem, De Gruyter, Berlin, New York, 1992.Google Scholar
  38. S. Osher and J.A. Sethian, Fronts propogating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys, 79 (1988), pp. 12–49.MathSciNetMATHCrossRefGoogle Scholar
  39. M. Paolini and C. Verdi, Asymptotic and numerical analyses of the mean curvature now with space-dependent relaxation parameter, Asymptotic Analysis to appear 1992).Google Scholar
  40. R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. Lond. A, 422 (1989), pp. 261–278.MathSciNetMATHCrossRefGoogle Scholar
  41. J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, Submitted to IMA J. Appl. Math (1991).Google Scholar
  42. J. Rubinstein, P. Sternberg and J.B. Keller, Fast reaction, slow diffusion and curve shortening, SIAM J. Appl. Math., 49 (1989), pp. 116–133.MathSciNetMATHCrossRefGoogle Scholar
  43. J. A. Sethian, Numerical algorithms for propogating interfaces: Hamilton-Jacobi equations and conservation laws, J. Diff. Geom., 31 (1990), pp. 131–161.MathSciNetMATHGoogle Scholar
  44. J. A. Sethian,’ The collapse of a dumbbell moving under its mean curvature’ in Geometric analysis and computer graphics, Springer-Verlag, New York, 1991, pp. 159–168.CrossRefGoogle Scholar
  45. H. M. Soner and P.E. Souganidis, Uniqueness and singularities of cylindrically symmetric surfaces moving by mean curvature, Preprint (1991).Google Scholar
  46. A. Stefan problem with surface tension, report of I.A.N, of C.N.R. Pavia (1984).Google Scholar
  47. A. Vlslntin, Surface tension effects in phase transitions, In ‘Materials Instabilities in Continuum Mechanics’ (ed J.M. Ball), Clarendon, Oxford, 1988a.Google Scholar
  48. A. Vlslntin, Stefan Problem with surface tension effects, In Mathematical Models for Phase Change Problems’ (ed. J.F. Rodrigues), Birkhäuser, Basel, 1988b.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • J. F. Blowey
    • 1
  • C. M. Elliott
    • 1
  1. 1.Mathematics DivisionUniversity of SussexBrightonUK

Personalised recommendations