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Motion by curvature in generalized Cahn-Allen models

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Abstract

The Cahn-Allen model for the motion of phase-antiphase boundaries is generalized to account for nonlinearities in the kinetic coefficient (relaxation velocity) and the coefficient of the gradient free energy. The resulting equation is

$$\varepsilon ^2 u_l = \alpha (u)(\varepsilon [\kappa (u)]^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \nabla \cdot \{ [\kappa (u)]^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \nabla u\} - f(u))$$

wheref is bistable. Hereu is an order parameter and κ and α are physical quantities associated with the system's free energy and relaxation speed, respectively. Grain boundaries, away from triple junctions, are modeled by solutions with internal layers when ε≪1. The classical motion-by-curvature law for solution layers, well known when κ and α are constant, is shown by formal asymptotic analysis to be unchanged in form under this generalization, the only difference being in the value of the coefficient entering into the relation. The analysis is extended to the case when the relaxation time for the process vanishes for a set of values ofu. Then α is infinite for those values.

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References

  1. S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to aniphase domain coarsening,Acta Metall. 27:1085–1095 (1979).

    Article  Google Scholar 

  2. G. Caginalp and P. C. Fife, Dynamics of layered interfaces arising from phase boundaries,SIAM J. Appl. Math. 48:506–518 (1988).

    Article  Google Scholar 

  3. J. W. Cahn and S. M. Allen, A microscopic theory for domain wall motion and its experimental verification in Fe−Al domain growth kinetics,J. Phys. (Paris)38(Coll. C7):51–54 (1977).

    Google Scholar 

  4. J. W. Cahn, C. M. Elliott, and A. Novick-Cohen, The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature, preprint.

  5. J. W. Cahn and J. E. Taylor, Surface motion by surface diffusion,Acta Met. Mat. 42:1045–1063 (1994).

    Article  Google Scholar 

  6. Xinfu Chen, Generation and propagation of interface in reaction-diffusion equations,J. Diff. Equations 96:116–141 (1992).

    Article  Google Scholar 

  7. L. C. Evans, H. M. Soner, and P. E. Souganidis, Phase transitions and generalized motion by mean curvature,Commun. Pure Appl. Math. 45:1097–1123 (1992).

    Google Scholar 

  8. P. C. Fife, Dynamics of internal layers and diffusive interfaces, inCBMS-NSF Regional Conference Series in Applied Mathematics, No. 53 (Society for Industrial and Applied Mathematics, Philadelphia, 1988).

    Google Scholar 

  9. P. C. Fife, Models for phase separation and their mathematics, inNonlinear Partial Differential Equations with Applications to Patterns, Waves, and Interface, M. Mimura and T. Nishida, eds. (KTK, Tokyo, 1994), pp. 153–212.

    Google Scholar 

  10. P. C. Fife and O. Penrose, Interfacial dynamics for thermodynamically consistent phasefield models with nonconserved order parameter, preprint.

  11. P. de Mottoni and M. Schatzman, Evolution geometrique d'interfaces,C. R. Acad. Paris 309:453–458 (1989).

    Google Scholar 

  12. P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces,Trans. Am. Math. Soc., in press.

  13. J. Rubinstein, P. Sternberg, and J. B. Keller, Fast reaction, slow diffusion and curve shortening,SIAM J. Appl. Math. 49:116–133 (1989).

    Article  Google Scholar 

  14. H. M. Soner, Ginzburg-Landau equation and motion by mean curvature, I: Convergence,J. Geom. Anal., in press.

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Authors and Affiliations

  1. Mathematics Department, University of Utah, Salt Lake City, Utah

    Paul C. Fife

  2. Mathematics Department, Heriot-Watt University, Edinburgh, Scotland

    Andrew A. Lacey

Authors
  1. Paul C. Fife
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  2. Andrew A. Lacey
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Dedicated to Oliver Penrose on the occasion of his 65th birthday.

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Fife, P.C., Lacey, A.A. Motion by curvature in generalized Cahn-Allen models. J Stat Phys 77, 173–181 (1994). https://doi.org/10.1007/BF02186837

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